monogenic wavelet frames for image analysis · only adaptable for image analysis by a tensor...

Technische Universität MünchenFakultät für Mathematik

Monogenic Wavelet Framesfor Image Analysis

Stefan Held

Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Universität Münchenzur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften (Dr. rer. nat.)

genehmigten Dissertation.

Vorsitzender: Univ.-Prof. Dr. Martin BrokatePrüfer der Dissertation: 1. Univ.-Prof. Dr. Brigitte Forster-Heinlein

2. Univ.-Prof. Dr. Uwe Kähler, Universitário de Santiago / Portugal3. Univ.-Prof. Dr. Rupert Lasser

Die Dissertation wurde am 12. Juli 2012 bei der Technischen Universität München eingereichtund durch die Fakultät für Mathematik am 22.November 2012 angenommen.

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Abstract

The subject of this thesis is the implementation of the Riesz transform via wavelet frames. Thisleads to an extension of the analytical wavelets, called monogenic wavelets, which allow todecompose an image into amplitude, phase, and a phase direction.

The reasons to use Riesz transforms for image analysis are the phase amplitude decomposi-tion they achieve via monogenic wavelet frames and the steerability of the resulting filters.Wavelets are the best choice for the implementation because they yield an amplitude phasedecomposition localized with respect to space and scale. Furthermore, wavelets are especiallywell adapted to Riesz transforms, since their generating operators commute with Riesz trans-forms.

We start with an investigation of the Riesz transform and present a novel definition of Riesztransforms of distributions, which results in an extension of the Bedrosian identity.

This is followed by a chapter on Clifford algebras – the language in which many of the followingresults are formulated. After a short introduction on Clifford algebra we introduce Clifford-Hilbert modules to state new results on left linear operators on Clifford-Hilbert modules.

These results are then used to define our Riesz transform as a steerable, left-linear, self-adjoint,unitary operator on a Clifford-Hilbert module. From this operator we define the monogenicsignal as extension of the analytical signal to higher dimensions. The monogenic signal yieldsa decomposition of a signal into amplitude, phase, and a phase direction, which gives the pre-ferred direction of the signal.

Using a new frame concept on Clifford-Hilbert modules – Clifford frames – we proof that due toits unitarity the Riesz transform maps frames onto Clifford frames. As a last step towards ourgoal we find suitable (i.e. radial) wavelet frames and implement them using the fast Fouriertransform.

The result is a steerable multiwavelet frame in arbitrary dimensions that yields a decomposi-tion into amplitude, phase and a direction which is local with respect to location and scale.Two applications are presented to show the usefulness of the constructed wavelets.

Then we study two tools, which can be used to improve filter design. The first one is the in-fluence of the Riesz transform on the Weyl-Heisenberg uncertainty relation. This allows tostudy the joint space-time localization of filters and their Riesz transforms. The second one isa sufficient condition for the Gibbs phenomenon to occur for wavelet frames.

The operator resulting from successively applying several partial Riesz transforms is called ahigher Riesz transform. We formulate higher Riesz transforms which are steerable unitary leftlinear operators on Clifford-Hilbert modules, and use the theory we developed earlier for theRiesz transform to implement the higher Riesz transforms. The higher Riesz transforms giverise to a higher monogenic signal, which yields a decomposition into amplitude, phase, andgeometrical information. Finally we explain how the steerable filters devised by Freeman andAdelson in [18] can be viewed as higher Riesz transforms applied to radial filters.

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Für meine Frau Anja, und meine Töchter Freya und Merle.

Mein Dank gilt:

Prof. Dr. Brigitte Forster, für Ihre Unterstützung und Förderung.

Prof. Dr. Rupert Lasser, der mit seinen Vorlesungen mein Verständniss von derAnalysis geprägt hat.

Prof. Dr. Brokate für seine Gastfreundschaft am Lehrstuhl M6.Peter Massopust der stets ein offenes Ohr für Fragen hatte und fürs Lesen derArbeit.

Contents

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contents of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Steerable filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.3 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.4 Basic operators of harmonic analysis . . . . . . . . . . . . . . . . . . . . . . 11

1.3.5 Wavelets and frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.6 Representations of groups and algebras . . . . . . . . . . . . . . . . . . . . . 14

1.4 Analytical signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.1 The Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.2 Analytical signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Riesz transforms 19

2.1 Riesz transforms and conjugate harmonic functions . . . . . . . . . . . . . . . . . 19

2.1.1 Riesz transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.2 Conjugate harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.3 Steerabilty of the Riesz transform . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Riesz transforms of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Preliminaries: Lizorkin spaces and derivatives of the Riesz multiplier . . . 27

2.2.2 Derivatives of the Fourier multiplier of the Riesz transform . . . . . . . . . 28

2.2.3 Riesz transforms of distributions modulo polynomials . . . . . . . . . . . . 30

2.2.4 Riesz transforms of certain tempered distributions . . . . . . . . . . . . . . 32

2.2.5 Hilbert and Riesz transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.6 The Bedrosian identity and Riesz transforms . . . . . . . . . . . . . . . . . . 36

3 Clifford algebras 41

3.1 Clifford algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Definition of Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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viii CONTENTS

3.1.2 Properties of Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.3 Zero divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.4 Paravectors and the paravector group . . . . . . . . . . . . . . . . . . . . . . 43

3.1.5 Scalar product, Euclidean measure and conjugation . . . . . . . . . . . . . 43

3.1.6 Complex Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Hypercomplex functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Clifford modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.2 Clifford-Hilbert modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.3 Operators on Clifford-Hilbert modules . . . . . . . . . . . . . . . . . . . . . 52

3.2.4 The Fourier transform on Lp (Rn )n . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Clifford analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Monogenic signals 65

4.1 Hypercomplex Riesz transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Differential operators and Riesz transforms . . . . . . . . . . . . . . . . . . . . . . 67

4.3 The monogenic signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 The phase of a monogenic signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Monogenic signals and Cauchy transforms . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Alternative analytical signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Monogenic wavelets 77

5.1 Introduction and example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Frames on Clifford-Hilbert modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Bessel sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 Clifford frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Monogenic wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.1 Monogenic wavelet frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Decay of monogenic wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5 Explicit construction of wavelet frames in L2(Rn ) . . . . . . . . . . . . . . . . . . . 91

5.5.1 Wavelets with arbitrary dilation matrices . . . . . . . . . . . . . . . . . . . . 91

5.5.2 Construction of regular partitions of unity . . . . . . . . . . . . . . . . . . . 94

5.5.3 An example of a tight wavelet frame . . . . . . . . . . . . . . . . . . . . . . . 95

5.5.4 An alternative construction for Riesz partitions of unity . . . . . . . . . . . 96

5.5.5 Interpretation via refinable functions . . . . . . . . . . . . . . . . . . . . . . 98

5.6 Implementation and application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.6.1 Implementational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Uncertainty relations 111

CONTENTS ix

6.1 The Heisenberg algebra and the Weyl Heisenberg group . . . . . . . . . . . . . . . 112

6.2 Uncertainty relations in one dimension . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.1 The Heisenberg uncertainty and the Hilbert transform . . . . . . . . . . . . 114

6.3 Uncertainty in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.1 A single uncertainty relation for sets of self adjoint operators . . . . . . . . 116

6.3.2 Uncertainty relations in Clifford-Hilbert modules . . . . . . . . . . . . . . . 117

6.3.3 The Weyl-Heisenberg uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 123

6.3.4 The Riesz transform and Heisenberg uncertainty . . . . . . . . . . . . . . . 125

6.3.5 The affine uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4 Discussion and comparison to the literature . . . . . . . . . . . . . . . . . . . . . . 129

7 The Gibbs phenomenon 131

7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2 The Gibbs phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3 Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 Higher Riesz transforms 137

8.1 Definitions and basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.1.1 Definition of spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . 138

8.1.2 Properties of spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . 138

8.1.3 The spaces Hk (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.2 Higher Riesz transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.2.1 Representations of the rotation group . . . . . . . . . . . . . . . . . . . . . . 143

8.2.2 Frames and higher Riesz transforms . . . . . . . . . . . . . . . . . . . . . . . 145

8.2.3 A phase concept for higher Riesz transforms . . . . . . . . . . . . . . . . . . 147

8.2.4 Composition of higher Riesz transforms . . . . . . . . . . . . . . . . . . . . 152

8.3 Implementational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.3.1 Explicit constructions for L2(R2) and L2(R3) . . . . . . . . . . . . . . . . . 156

8.3.2 Implementational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.3.4 Comparison with other approaches . . . . . . . . . . . . . . . . . . . . . . . 161

A The Appendix 165

A.1 Conventions and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.1.1 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.1.2 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.1.3 Spaces of Lebesque measurable funtions . . . . . . . . . . . . . . . . . . . . 166

A.1.4 Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.1.5 The formula of Faá di Bruno . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.1.6 Multiresolution Analysis (MRA) . . . . . . . . . . . . . . . . . . . . . . . . . 167

x CONTENTS

A.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A.2.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A.2.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A.2.3 The direct product of distributions . . . . . . . . . . . . . . . . . . . . . . . . 170

A.2.4 Convolution of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A.2.5 Products and distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

A.2.6 Fourier transforms of distributions . . . . . . . . . . . . . . . . . . . . . . . 172

A.2.7 A Paley-Wiener theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A.2.8 The Fourier transform and convolution of distributions . . . . . . . . . . . 173

List of Figures 177

Bibliography 179

Index 183

Chapter 1

Introduction

1.1 Introduction

This thesis centers around two ideas that are important in signal and image processing.

Phase information is a valuable tool in signal processing, where it is acquired using analyt-ical wavelets. (For details on analytical wavelets see [43] and the references therein.) Phaseinformation is desirable for image analysis since it allows to investigate image features inde-pendent of illumination. However, analytical wavelets are only one dimensional and thereforeonly adaptable for image analysis by a tensor product approach.

Steerable filters are directional filters. Directional filters are used in image processing for ex-ample in texture analysis, edge detection, image data compression, motion analysis and imageenhancement. (See [18] for references.) For these tasks filters with arbitrary directions are of-ten required. A steerable filter in an arbitrary direction is synthesized as a linear combinationof a set of basis filters [18] - in contrast to other directional filters like Gabor or Morlet wavelets,where a separate filter is necessary for each orientation. Of course this restricts the number ofpossible directions for non steerable directional filters. However, steerable filters are usuallyderived as steerable pyramids that yield only discrete filters.

Our aim is to transfer the concept of analytic wavelets into arbitrary higher dimensions usingthe Riesz transform which generates steerable filters.

Our approach is new in that we use a wavelet frame construction that works in n-D for anyn ∈N. Furthermore this approach is intrinsically continuous while still allowing for an effi-cient discrete implementation. The benefit of our approach is that we retrieve phase infor-mation which is localized in space and scale from a signal of arbitrary dimension using filterswhich are steerable multiwavelets. The phase we derive in this multi-scale approach is betterinterpretable than the one derived by the monogenic signal, because it gives phase informa-tion on all scales separately, whereas, in contrast, the phase derived from the monogenic signalis an average over all scales.

The fact that we use wavelets as filters enables us to use the advantageous properties of contin-uous wavelets to analyze the properties of our filters - continuity, decay, vanishing moments

1

2 CHAPTER 1. INTRODUCTION

and localization carry information about important properties as for example the approxima-tion order.

We noted that the Riesz transforms of our wavelets are steerable filters. Indeed they can beviewed as steerable pyramids. But this connection is not one way – using higher Riesz trans-forms we state a method to find the operators which generate steerable filters. In this waysteerable filters are freed of the discrete setting of steerable pyramids and can be viewed assteerable operators which may act on radial wavelet frames to generate steerable wavelets aswell as on any radial filters that generate a frame.

To show that the Riesz transform of a wavelet frame yields a multiwavelet frame and to in-vestigate the space that monogenic wavelets generate we consider Clifford frames on non-commutative Clifford-Hilbert modules. Since frames and multiwavelet frames can be viewedas Clifford frames this setting allows us to proof that the Riesz transform preserves Cliffordframes. The same means enable us to study the space generated by monogenic wavelet frames.

More interesting than a historical account of this field is an account of the development in thisexciting area of research – driven by several groups of strong researchers – that occured duringthe time in which this thesis was written. (An account of the earlier development of this fieldis given in our paper [24]. Especially the works of Elias Stein and Guido Weiss [45, 46] and thework of Michael Felsberg[15] should be noted in this context.)

In 2009 Michael Unser and Dimitri Van De Ville published a paper [52] in which they demon-strated an implementation of the Riesz transform in 2-D using an orthogonal basis of approxi-mately radial wavelets. For the proof that the Riesz transform of an wavelet frame is once againa wavelet frame this paper refers to our paper [23] which was at that time in review process.

In [38] Juan Romero et. al. gave a method to construct radial tight wavelet frames and gaveexamples of radial wavelet frames with |det A| radial mother wavelets, where A is the dilationmatrix.

A first part of this thesis was published in [23] and [24] which contains the implementationof the Riesz transforms in arbitrary dimension using tight wavelet frames with a single radialmother wavelet and the theory of Clifford frames to proof the frame property of the Riesz trans-forms.

These papers contain several new contributions:In contrast to Michael Unser, we use wavelet frames in arbitrary dimensions. These frames areperfectly radial, which improves the quality of the derived phase and allows for proper steer-ability.In contrast to Romero et. al. we use a tight wavelet frame with exactly one mother wavelet.This is less redundant and allows for an easier application of the Riesz transform. Further-more, we proof that the Riesz transform maps wavelet frames to multiwavelet frames. It is thisproperty that makes it possible to design perfect reconstruction filter banks. In several otherpublications in this field e.g. [42, 51, 53, 4, 48, 49] our article [24] was cited.

In [24] we proposed to study higher Riesz transforms to generate steerable wavelets.

A first step in this direction were the papers [51] and [53] where Michael Unser et. al. in-troduced higher order Riesz transforms, which correspond to a composition of higher Riesztransforms we propose in this thesis.

The chapter on higher Riesz transforms in this thesis is concerned with giving a comprehensivetheoretical background on higher Riesz transforms and steerable filters. Using this backgroundwe can discern the minimal higher Riesz transforms which are steerable and give the meansto combine these adaptively to higher Riesz transforms with more interesting geometrical fea-tures. This is important to gain flexibility while limiting redundancy. The number of partial

1.2. CONTENTS OF THE CHAPTERS 3

higher Riesz transforms linearly increases the runtime and memory use of the algorithms.

Another advantage of our approach is that we can use our theoretic knowledge – namely thatthe steerability is a manifestation of a representation of the rotation group – to gain insightsinto the higher order Riesz transforms of Michael Unser’s. We show that these higher orderRiesz transforms come in two sets – the filters of odd and even order – in which all sets of higherRiesz transforms of lower order are contained in those of higher order. As a consequence theredundancy of these higher order Riesz transforms rapidly increases as the order increases.

Furthermore we were able to show a connection between higher Riesz transforms of waveletsand the steerable pyramids described in [18]. It is easily seen that the implementation of thehigher Riesz transform of a wavelet frame is a steerable pyramid. On the other hand we couldshow that steerable pyramids correspond to higher Riesz transforms applied to filters whichform a perfect reconstruction filter bank.

1.2 Contents of the chapters

Chapter 1 provides a short summary of basic facts on harmonic analysis and analytical sig-nals which will prove useful in the remainder of this thesis.

Chapter 2 introduces the Riesz transform. The first part of this chapter section 2.1 intro-duces the Riesz transform as a unique extension of the Hilbert transform, states the most im-portant classical results on the Riesz transform, gives an interpretation of the term “analytical”in the name “analytical signal” and states that Riesz transforms are steerable.

Chapter 2Riesz transforms

Chapter 7Gibbs phenomenon

Chapter 4Monogenic signals

Chapter 5Monogenic Wavelets

Chapter 8Higher Riesz

transforms

Chapter 3Clifford algebras

Chapter 6Uncertainty relations

Figure 1.1: Interrelation of the chapters

The second part of this chapter, section 2.2, presents new results. We define the Riesz trans-form for two classes of distributions – distributions modulo polynomials (Definition 2.2.10)and tempered distributions whose support in the Fourier domain does not contain the ori-gin (Theorem 2.2.13). As a application we state a novel extension of the Bedrosian identity toHilbert transforms of distributions (Theorem 2.2.18). Our definition of the Riesz transform ofdistributions is especially well suited for this task — indeed the prerequisites for the Bedrosianidentity are all that is required for the Hilbert transform of the involved distributions to be welldefined. By means of a certain new operator that maps one dimensional tempered distribu-tions to n dimensional tempered distributions (Proposition 2.2.14) we state a new connectionbetween the Hilbert and the Riesz transform (Theorem 2.2.15). From this connection we thenobtain a Bedrosian identity for Riesz transforms of certain distributions (Theorem 2.2.19).


Chapter 3 introduces Clifford algebras and Clifford Hilbert-modules — the language inwhich many of the following results are formulated.

Section 3.1 gives a short introduction to Clifford algebra.

Section 3.2 introduces Clifford modules to provide the setting in which we then introduce somealready known facts about functional analysis in Clifford-Hilbert modules.

Section 3.2.3 considers for the first time systematically left-linear operators on Clifford-Hilbertmodules. This consideration results in new insights that are stated in Theorem 3.2.7 – Corol-lary 3.2.13. These insights are essential for the definition of the monogenic signal in chapter 4,for deriving uncertainty relations in chapter 6 and for the development of Clifford frames insection 5.2.

The final part of the chapter treats some basics of Clifford analysis which extends classicalcomplex analysis to higher dimensions. The monogenic signal is closely connected to Cliffordanalysis as revealed in chapter 4.

In chapter 4 we formulate the monogenic signal as an extension of the analytical signal tohigher dimensions and examine the properties of this monogenic signal.

Section 4.1 establishes the hypercomplex Riesz transform on which the monogenic signal isbased. The first step is the definition of the hypercomplex Riesz transform R in Definition 4.1.1that extends the Hilbert transform iH . What is novel about this definition is that we definethe Riesz transform as a left linear-operator on a Clifford-Hilbert module. As a consequencethe (hypercomplex) Riesz transform we define is unitary and self-adjoint as is shown in Theo-rem 4.1.4. It follows that the hypercomplex Riesz transform is its own inverse.

Section 4.2 shows that the hypercomplex Riesz transform is closely connected to differentialoperators. This novel result is stated in Theorem 4.2.1.

We define the monogenic signal in section 4.3.

In section 4.4 we state the decomposition of the monogenic signal into phase, amplitude, andphase direction. The later provides the preferred direction of a signal. From the phase wederive an instantaneous frequency for the monogenic signal.

By the properties of the Riesz transforms defined in Definition 4.1.1 it follows that the mono-genic signal is generated by a projection M : f 7→ fm (Theorem 4.3.3) that is closely connectedto Clifford analysis. We state these connections in section 4.5.

The final section 4.6 shows that the monogenic signal is uniquely defined as an extension ofthe analytical signal by the condition that it is compatible with rotations – which generate thesymmetry group we have to deal with in image analysis.

Some of the results of this chapter have been published in the peer reviewed article [24] and in[23].

The agenda of chapter 5 is to implement the monogenic signal – given by a hypercomplexsingular integral operator - via multi-wavelet frames which amounts to a multiplication with aset of explicitly known filters in the Fourier domain.

This is the central chapter of this thesis - all previous chapters set the background for what weachieve in this chapter and all further chapters extend what we do in this chapter.

The first section 5.1 explains why wavelet frames are the tool of choice for the implementationof the monogenic signal. Example 5.1.2 illustrates our approach using the example of analyti-cal wavelet frames and sets the roadmap for sections 2 and 3.

1.2. CONTENTS OF THE CHAPTERS 5

Section 5.2 extends the concept of frames on Hilbert spaces to the setting of Clifford-Hilbertmodules – the space in which the monogenic signal lives – introduced in section 3.2. Theconcept of Clifford frames – to use Clifford algebra valued frame coefficients to achieve a framedecomposition – appears to be entirely new.

Section 5.3 defines the notion of a hypercomplex wavelet transform which is compatible withClifford frames (Definition 5.3.1) and the monogenic signal. Theorem 5.3.3 shows that we canderive a monogenic wavelet frame from a standard wavelet frame for L2(Rn) using a surjectivehypercomplex operator.

In section 5.4 we state that under mild conditions the decay rate of the wavelet frames is pre-served by the Riesz transform.

Section 5.5 is dedicated to the search for suitable wavelet frames. First, we introduce resultswhich reduce the problem of constructing wavelet frames to the problem of constructing Rieszpartitions of unity. Theorem 5.5.7 provides a new way to find such Riesz partitions of unity. Anexplicit construction is then given in Example 5.5.1. Theorem 5.5.8 shows how to derive Rieszpartitions of unity from agiven compactly supported wavelet orthonormal bases of L2(R).

Finally, section 5.6 presents the implementation of the wavelet frames of Example 5.5.1 as animageJ plugin. The software and the applications have been the topic of the diploma thesis ofMartin Storath [47] under supervision of the author.

The result is a tight steerable multiwavelet frame in arbitrary dimensions that yields a decom-position into amplitude, phase and a direction which is local with respect to location and scale.

The tight multi-wavelet frames do have a simple and explicit form in the Fourier domain,whence all algorithms are implemented in the frequency domain. The compact support ofthe wavelet frames in the Fourier domain allows a fast filter-bank implementation with perfectreconstruction property. The Shannon sampling theorem ensures loss-less down-samplingafter each filter step.


In chapter 6 we examine uncertainty relations for the Riesz transforms. Uncertainty rela-tions quantify the joint localisation in time or space domain and Fourier domain. For that rea-son they are an important tool in the design of wavelets. In chapter 5 we implement the Hilbertand the Riesz transform via wavelets. Therefore it will be the topic of chapter 6 to compute theeffect of the Hilbert and the Riesz transform on the Weyl-Heisenberg uncertainty relation.

In Theorem 6.2.3 we prove the new result that under mild assumptions the Weyl-Heisenberguncertainty relation is invariant under the Hilbert transform.

Since the Riesz transform R is unitary whereas the partial Riesz transforms Rα are not we con-sider a single uncertainty relation for the Riesz transform rather than a set of uncertainty re-lations for the partial Riesz transforms. This requires a new kind of uncertainty relation forvector valued functions.

In section 6.3 we pursue this goal along two different paths: a classical approach Theorem 6.3.2based on the Cauchy-Schwartz inequality and an uncertainty relation on Clifford-Hilbert mod-ules Theorem 6.3.3. Both paths yield novel uncertainty relations.

In contrast to the set of classical Weyl-Heisenberg uncertainty relations in higher dimensionsstated in Corollary 6.3.1 the Weyl-Heisenberg uncertainty relation we derive in Theorem 6.3.10is invariant under rotations. This property of invariance with respect to rotation makes themeasily interpretable in the context of image processing.


In Theorem 6.3.12 we use our results to derive a novel Weyl-Heisenberg uncertainty relation forthe Riesz transform and to compute the effect of the Riesz transform on the Weyl-Heisenberguncertainty relation Theorem 6.3.10.

The Chapter will end with a new affine uncertainty relation Theorem 6.3.13 as a further exam-ple for an application of Theorem 6.3.3.

In chapter 7 we state sufficient conditions for the Gibbs phenomenon to occur in a waveletframe decomposition. The Gibbs phenomenon describes the appearance of over- and under-shoots of a wavelet approximation near a jump discontinuity. We examine the existence of theGibbs phenomenon for wavelet frames which are based on a certain kind of multiresolutionanalysis, i.e., for which a suitable scaling function exists.

Chapter 8 is dedicated to the study of higher Riesz transforms. The operator resulting fromsuccessively applying several partial Riesz transforms is called a higher Riesz transform.

The steerability of the Riesz transform corresponds to a unitary representation of the rotationgroup on the set of Riesz transformed functions. In section 8.2 we show that for n > 1 there arean infinite number of such unitary representations of the rotation group based on the sphericalharmonics. From these we derive steerable unitary left linear operators on Clifford-Hilbertmodules - the higher Riesz transforms. These higher Riesz transforms are the smallest possiblesets of partial higher Riesz transforms which are invertible and steerable.

Combining these minimal higher Riesz transforms in Corollary 8.2.18 we derive higher Riesztransforms which are geometrically richer at the cost of higher redundancy.

We use the theory of Clifford frames and hypercomplex wavelets we developed for the Riesztransform in chapter 5 to implement the higher Riesz transforms. The higher Riesz transformsgive rise to a higher monogenic signal, which yields a decomposition into amplitude, phase,and geometrical information. Theorem 8.2.15 states the connection of the higher monogenicsignal to certain generalized Cauchy Riemann equations derived from higher Dirac operatorsof higher order which are the square-root of a power of the Laplace operator.

In Remark 8.2.19 we learn how to construct higher Riesz transforms which contain the direc-tional information of higher derivatives. This approach is similar to the steerable pyramids ofFreeman and Adelson in [18] – indeed in Remark 8.2.22 we show that these steerable pyramidshave a close connection to higher Riesz transforms.

In section 8.3 we give explicit constructions of higher Riesz transforms for the interesting casesn = 2 and n = 3.

The Appendix A states some conventions and give some further details on distributions.

1.3. PRELIMINARIES 7

1.3 Preliminaries

In this section we will state some basic facts of harmonic analysis that we often will use lateron.

1.3.1 Steerable filters

Steerable filters were introduced by Freeman and Adelson in [18]. A steerable filter is a functionwhich can be steered, i.e. arbitrary rotated versions of the function can be computed by a linearcombination of a certain finite set of rotated versions of the function.

R1 sin(3π/16)R1 +cos(3π/16)R2

2−1/2(R1 + R2) R2

Figure 1.2: Example of steerable filters: the basis filters are R1,R2 the linear combinations arerotated versions of the filters.

Definition 1.3.1 (Steerable filter)LET faa∈Sn−1 be a set of directed filters.

THEN fa is called a steerable filter, iff the steerable filter with respect to an arbitrary directiona ∈ Sn−1 can be written as a linear combination of filters with respect to a set of basis directionsal l=1,...,k ,. That is for every a ∈ Sn−1 there exists a vector va = (va,1, . . . , va,l ) ∈Rk such that

fa =k∑

l=1va,l fal .


1.3.2 Tempered distributions

We will start with some basic facts about tempered distributions. A more thoroughly introduc-tion is given in the appendix, section A.2.

Definition 1.3.2 (Schwartz Class)LET f ∈ C∞(Rn), n ∈N. Furthermore let α = (α1, . . . ,αn) ∈Nn

0 be a multi-index. As usual wewrite

∂α f (x) = ∂|α|

∂α1 x1 · · ·∂αn xnf (x),

where |α| =∑nl=1αl . Let

ρ j ,k ( f ) = supx∈Rn , |α|≤ j

(1+|x|)k |∂α f (x)|.

THEN the Schwartz class of rapidly decreasing functions S (Rn) is defined as

S (Rn) := f ∈C∞(Rn) : ∀ j ,k ∈N0∃C j ,k ∈R+ : ρ j ,k ( f ) =C j ,k <∞.

Theorem 1.3.3S (Rn) is a Fréchet space with the set of seminorms ρ j ,k .

S (Rn) is a dense subspace of Lp (Rn), 1 ≤ p <∞.

Proof. For a proof see [37] V.3.

Definition 1.3.4 (Tempered distributions)Elements of the dual space S ′(Rn) of S (Rn) are called tempered distributions.

The support of f ∈S ′(Rn) is defined by

supp( f ) :=⋂K ⊂Rn compact :φ ∈S (Rn), supp(φ) ⊆ K c ⇒ f (φ) = 0.

Definition 1.3.5 (Derivatives of distributions)LET f ∈S ′(Rn) and let α ∈Nn

0 .

THEN the derivative ∂α f ∈S ′(Rn) is defined as

∂α f (φ) := (−1)|α| f (∂αφ), ∀φ ∈S (Rn).

Corollary 1.3.6The derivative of a distribution

∂α : S ′(Rn) →S ′(Rn)

as given in Definition 1.3.5 is well defined.

Proof. For a proof see for example [41] 6.12.

Definition 1.3.7 (The direct product of distributions)LET n,m ∈N, f ∈S ′(Rn), and g ∈S ′(Rm).

THEN the direct product f × g ∈S ′(Rn+m) of f and g is defined as

( f × g )(φ) := f (g (φ)) = g ( f (φ)), ∀φ ∈S (Rn ×Rm).


Definition 1.3.8 (Convolution of distributions)LET f , g ∈S ′(Rn).

THEN the convolution f ∗ g ∈S ′(Rn) of f and g is defined by

f ∗ g (φ) = f × g(φ( · f + · g )

), ∀φ ∈S (Rn).

Remark 1.3.9 (Well-definedness of direct product and convolution)The direct product of distributions given in Definition 1.3.7 is well-defined and commutative.(See Theorem A.2.7.) The convolution in Definition 1.3.8 is well defined if at least one of thetempered distributions has compact support. A proof of this fact can be found in the appendixTheorem A.2.9.

Theorem 1.3.10 (Product of distributions)LET g ∈S ′(Rn) and let f ∈C∞(Rn) such that

∀α= (α1, . . . ,αn) ∈Nn0 ∃kα ∈N0,Cα > 0 : |(∂α f )(x)| ≤Cα(1+|x|)kα , ∀x ∈Rn .

THEN the product f g ∈S ′(Rn) is well defined by

f g (φ) = g ( f φ), ∀φ ∈S (Rn).

1.3.3 The Fourier transform

Definition 1.3.11 (Fourier transform)We define the Fourier transform of a function f ∈S (Rn) by

F ( f )(ξ) = f (ξ) :=∫Rn

f (x)e−2πi x·ξd x =∫Rn

f (x)e−2πi ⟨x,ξ⟩Rn d x, f.a.a. ξ ∈Rn .

Theorem 1.3.12The Fourier transform is an isomorphism on S (Rn). Its inverse is given by

F−1( f )(x) :=∫Rn

f (ξ)e2πi x·ξdξ, f.a.a. x ∈Rn .

The Fourier transform of a function in Lp (Rn), 1 ≤ p <∞, is defined by extension of this def-inition to Lp (Rn) and denoted by F . By the Plancherel theorem the Fourier transform is anisometry on L2(Rn).

The Fourier transform F : S ′(Rn) →S ′(Rn) of a tempered distribution f ∈S ′(Rn) is definedas f (φ) = f (φ), ∀φ ∈S (Rn). It is an isomorphism on S ′(Rn).

Proof. For the proof see [40], Chapter 7, and Theorem 7.4-1in [58] .

Theorem 1.3.13 (Plancherel)LET f ∈ L2(Rn).

THEN

‖ f ‖L2(Rn ) = ‖ f ‖L2(Rn ).

Proof. A proof can be found in [40].


Corollary 1.3.14 (Parsevals formula)LET f , g ∈ L2(Rn).

THEN

⟨ f , g ⟩ = ⟨ f , g ⟩.


Theorem 1.3.15 (Fourier transform and convolution)LET f , g ∈S ′(Rn) such that f has compact support.

THEN

f g = f ∗ g .

Proof. The proof is given in the appendix Theorem A.2.14.

Theorem 1.3.16 (Poisson’s Summation Formula)

(i) LET f :Rn →R such that

∃ε> 0, C > 0 : | f (x)| ≤C (1+|x|)−n−ε∧| f (ξ)| ≤C (1+|ξ|)−n−ε.

THEN ∑k∈Zn

f (x +k) = ∑k∈Zn

f (k)e2πi k·x ,

holds pointwise for all x ∈Rn , and both sums converge absolutely for all x ∈Rn .

(ii) LET f ∈ L2(Rn) such that∑

k∈Zn f (x +k) ∈ L2([0,1]n) and∑

k∈Zn | f (n)|2 <∞. THEN∑k∈Zn

f (x +k) = ∑k∈Zn

f (k)e2πi k·x ,

holds almost everywhere, and both sums converge in L2(Rn).

Proof. For a proof see [20] Chapter 1.4.

Theorem 1.3.17 (Fourier transforms and derivatives)LET f ∈S (Rn) and let α ∈Nn

0 .

THEN

F (∂α f )(ξ) = (2πiξ)α f (ξ), ∀ξ ∈Rn ,

andF

((−2πiξ)α f

)(ξ) = ∂α f (ξ).

Proof. For a proof see for example [20].

Definition 1.3.18 (The Laplace operator)LET f ∈S (Rn). The Laplace operator is given by

4 f (x) :=n∑α=1

∂2

∂x2α

f (x).

By Definition 1.3.5, it can be extended to f ∈S ′(Rn).


Definition 1.3.19 (Sobolev and Bessel potential spaces)LET k ∈N, p ∈ [1,∞[. The Sobolev spaces are defined as

W k,p (Rn) = f ∈S ′(Rn) : ∂α f ∈ Lp (Rn), ∀α ∈Nn

0 , |α| ≤ k.

Let s ∈R. The Bessel potential spaces are given by

W s (Rn) := f ∈S ′(Rn) :

∫Rn

f 2(ξ)(1+|ξ|2)s <∞.

Theorem 1.3.20LET f ∈S ′(Rn), k ∈N.

THEN ∂α f ∈ L2(Rn), ∀|α| ≤ k ⇔ ∫Rn | f (ξ)|2(1+|ξ|2)k <∞.

As a consequenceW k,2(Rn) =W k (Rn).

Proof. For a proof see [20], Lemma 1.2.3.

1.3.4 Basic operators of harmonic analysis

We will now define some basic operators of harmonic analysis.

Definition 1.3.21 (Translation, modulation, dilation and rotation)LET t ,b ∈Rn , d ,di ∈R, ∀1 ≤ i ≤ n, i ∈N and f :Rn 7→C.

THEN we define the translation operator by

Tt f (x) := f (x − t ),

and the modulation operator by

Mb f (x) := e2πi b·x f (x).

We define the operation of a rotation ρ ∈ SO(n) on a function f by

ρ f (x) = f (ρx).

We define a dilation by a matrix A with eigenvalues of modulus greater than one via

A f (x) := |det(A)|1/2 f (Ax).

This definition is usually too general so that we have to restrict to certain types of dilation.

The isotropic dilation is defined by

Dd f (x) := |d |n/2 f (d x).

In wavelet theory this is often combined with a rotation ρ which gives what we will call rotateddilation

Dd f (x) := Ddρ f (x) = |d |n/2 f (dρx).


In product space settings where rotations are not considered, a non-isotropic dilation is usefulwhich will be defined as

Ddi ni=1

f (x) :=(

n∏i=1

di

)1/2

f (diag(d1, . . . ,dn)x),

where diag(d1, . . . ,dn) is the diagonal n ×n-matrix with entries d1, . . . ,dn ∈R.

Theorem 1.3.22 (Fourier transforms of translation, modulation, dilation and rotation)LET t ,b ∈Rn , d ,di ∈R, ∀1 ≤ i ≤ n, i ∈N and f :Rn 7→C. Let f ∈ Lp (Rn) or f ∈S (Rn).

THEN modulation and translation are dual with respect to the Fourier transform in the sensethat

Tt f = M−t f

and

Mb f = Tb f .

Both are unitary operators with respect to the L2-norm, which is clear for Mb and hence by thePlancherel theorem for Tt .

The inverse of the rotation operator is the operator corresponding to the transposed matrixwhich is the inverse of the rotation matrix and

F (ρ f ) = ρF ( f ), ∀ f ∈ L2(Rn).

The Fourier transform of the isotropic dilation isDd f = Dd−1 f = D−1d f , ∀ f ∈ Lp (Rn).

The Fourier transform of the rotated dilation isDd f = Ddρ f = Dd−1ρ f = (DT )−1 f , ∀ f ∈ Lp (Rn).

Proof. Let f ∈ Lp (Rn) or f ∈S (Rn). Modulation and translation are dual since

F (Tt f )(ξ) =∫Rn

f (x − t )e−2πi ⟨x,ξ⟩d x =∫Rn

f (x)e−2πi ⟨x+t ,ξ⟩d x = e2πi ⟨−t ,ξ⟩∫Rn

f (x)e−2πi ⟨x,ξ⟩d x

= M−t F ( f )(ξ)

and

F (Mb f )(ξ) =∫Rn

f (x)e2πi ⟨b,x⟩e−2πi ⟨x,ξ⟩d x =∫Rn

f (x)e−2πi ⟨x,ξ−b⟩d x

= TbF ( f )(ξ), ∀ξ ∈Rn .

All the other operators are of the form A f ( · ) := f (A · ), where A ∈ Gl(n) is an invertible matrix.

In general it holds that

F (A f )(ξ) =∫Rn

f (Ax)e−2πi ⟨x,ξ⟩d x =∫Rn

f (x)e−2πi ⟨A−1x,ξ⟩|det(A−1)|d x

=∫Rn

f (x)e−2πi ⟨x,(A−1)∗ξ⟩|det(A−1)|d x

= |det(A−1)|(A−1)∗F ( f )(ξ).


1.3.5 Wavelets and frames

Definition 1.3.23 (Wavelet transform)LET ψ ∈ L2(Rn) and let D be a dilation matrix.

THEN the set of dilated and translated versions of ψ

D j Tkψ j ,k

is called a wavelet system, where j ∈Z and k ∈Zn or j ∈R and k ∈Rn with mother waveletψ.

LET f ∈ L2(Rn).

THEN

Wψ( f ) := ⟨ f ,D j Tkψ⟩ j ,k

is called the wavelet transform of f .

Definition 1.3.24 (Frame)LET H be a Hilbert space, let I be a countable set and ψk k∈I ⊂ H.

THEN ψk k∈I is called a frame for H iff ∃0 < A ≤ B <∞ such that ∀ f ∈ H the frame inequality

A‖ f ‖2 ≤ ∑k∈I

|⟨ f ,ψk⟩|2 ≤ B‖ f ‖2

holds. In this case A and B are called the lower respectively the upper frame bound. ψk k∈I

is called a tight frame iff A = B.

Definition 1.3.25 (Wavelet frame)A wavelet system is called a wavelet frame if it is a frame.

Theorem 1.3.26 (Dual frame and frame decomposition)LET ψk k∈I ⊂ H be a frame for a Hilbert space H .

THEN there exists a frame φk k∈I ⊂ H called a dual frame such that ∀ f ∈ H the frame de-composition

f = ∑k∈I

⟨ f ,ψk⟩φk = ∑k∈I

⟨ f ,φk⟩ψk

holds.

Proof. For a proof and for further details on frame theory see [10].

The image of a frame under a surjective operator yields a frame:

Theorem 1.3.27 (Frames and operators)LET H ,G be Hilbert spaces. Let fk k∈N be a frame for H with bounds 0 < A ≤ B <∞ and letU : H →G be a bounded surjective operator.

THEN U fk k∈N is a frame for G with frame bounds A‖U †‖−2,B‖U‖2. (Here † denotes thepseudo-inverse.)

Proof. This is Theorem 5.3.2 in [10].


1.3.6 Representations of groups and algebras

Definition 1.3.28 (Representation)LET G be a group or an algebra and let X be a linear space. A representation of G in X is ahomomorphism R from G to the vector space L(X ) of linear, bijective mappings from X to X .Let g ,h ∈G and let e ∈G be the identity.

THEN

R(g )R(h) =R(g h),

R(e) = IdL(X ),

and in the case that G is an algebra, we have in addition, that

R(g +h) =R(g )+R(h).

A representation is called reducible if there exists a subspace V of X that is invariant under therepresentation R, i.e.,

R(g )v ∈V , ∀v ∈V , g ∈G .

Then the restriction of R to V is called a subrepresentation of R.

The factor representation is the representation on the factor space X /V naturally given by R.

A representation is called unitary if the space X is a Hilbert space and R(g ) is unitary for allg ∈G.

Remark 1.3.29LET R be a unitary reducible representation on a vector space X with invariant subspace V .

THEN R|V ⊥ is a unitary representation.

1.4. ANALYTICAL SIGNALS 15

1.4 Analytical signals

1.4.1 The Hilbert transform

We define analytical signals using the Hilbert transform, so we will first have a look at theHilbert transform itself.

Definition 1.4.1 (The Hilbert transform)LET f ∈ Lp (R), where 1 < p <∞.

THEN the Hilbert transformH : Lp (Rn) 7→ Lp (Rn),

is defined as the following Cauchy principal value:

H f (x) := limδ→0

1

π

∫|y |>δ

f (x − y)

yd y.

Theorem 1.4.2 (Properties of the Hilbert transform)LET f ∈ Lp (R), where 1 < p <∞.

THEN the Hilbert transform

(i) anti-commutes with reflection, i.e. H ( f (− · ))(x) =−H ( f )(−x);

(ii) commutes with translation;

(iii) commutes with dilation;

(iv) ∃A(p) such that ‖H f ‖p ≤ A(p)‖ f ‖p , ∀ f ∈ Lp (R). This result is called the theorem of M.Riesz;

(v) is uniquely defined by properties (i)-(iv) modulo a multiplicative real constant;

(vi) the corresponding Fourier multiplier is given by H f (x) =−i sgn(x) f (x);

(vii) H 2( f ) =− f .

Proof. ad (i)

H ( f (− · ))(x) = 1

πlimε→0

∫|t |>ε

f (t −x)

td t = 1

πlimε→0

∫|t |>ε

− f (−x − t )

td t =−H ( f )(−x).

ad (ii)

H (Tl f )(x) = 1

πlimε→0

∫|t |>ε

f (x − l − t )

td t =H ( f )(x − l ) = Tl H ( f )(x).

ad (iii)

H (Ds f )(x) = s−1/2 1

πlimε→0

∫|t |>ε

f (s(x − t ))

td t = s−1/2 1

πlimε→0

∫|t |>ε

f (sx −τ)

τ/s

1

sdτ

= DsH ( f )(x).

ad (iv) We will not proof this here. For a proof see [59], Chapter 3.


ad (v) We will proof this in a more general case in Theorem 2.1.2.

ad (vi) This will be proven in Theorem 2.1.2 in a more general setting.

ad (vii) For the Fourier transforms it is obviously true that

àH 2( f )(t ) =−i sgn(t )(−i sgn(t )) f (t ) =− f (t ).

1.4.2 Analytical signals

Definition 1.4.3 (The analytical signal)LET f ∈ Lp (R,R), 1 < p <∞ be a 1-D real-valued signal.

THEN the analytical signal fa is defined by

fa := f + iH f ∈ Lp (R,C).

Theorem 1.4.4 (Properties of analytical signals)LET f ∈ Lp , p ∈]1,∞[.

THEN

(i) F ( fa)(w) :=F ( f )(w)+ sgn(w)F ( f )(w), i.e., the Fourier transform of the analytical sig-nal vanishes for negative frequencies.

(ii) fa ∈ Lp (R,C) and f = Re( fa), i.e., the analytical signal is complex-valued and the real-valued signal is its real part.

(iii) An analytical signal can be decomposed into fa(t ) = | fa(t )|e iφ(t ), where a(t ) = | fa(t )| iscalled the amplitude andφ(t ) :R→ [0,2π[ the phase, which is uniquely defined by thisdecomposition.

Often the phase defined by the above decomposition is interpreted as the phase of a localFourier transform assuming that the behaviour under translation, modulation, and rotation isthe same as for the Fourier transform.

The use of the phase is inspired by a paper of Oppenheim and Lim [35], who claim that theimportant information of an image is in the phase of the Fourier transform.

However, it should be noted that this interpretation for the phase is somewhat questionablesince the phase acquired by the analytical signal is at best approximately equal to the phase of alocal Fourier transform and Oppenheim and Lim argue in the same paper that their argumentsdo not hold for a localised Fourier transform.

The above decomposition of the analytical signal implies that the signal itself may be uniquelydecomposed into amplitude and frequency part via

f (t ) = a(t )cos(φ(t )), (1.1)

where a(t ) is the amplitude of the analytical signal and φ(t ) is just the phase of the analyticalsignal.

Inspired by this decomposition is the definition of the analytic instanteous frequency of a sig-nal.

1.4. ANALYTICAL SIGNALS 17

Figure 1.3: Example of an analytical signal: f (x) = 4 sin(πx)x2+1

, H f (x) = 4 e−π−cos(πx)x2+1

.This example can be found in [30].

Figure 1.4: Phase of the function f .


Definition 1.4.5 (Instantaneous Frequency)LET f ∈ Lp (R), where 1 < p <∞.

THEN the instantaneous frequency ω(t ) of a signal f is defined on its support by

ω(t ) :=

H ( f )(t ) f ′(t )− f (t )H ( f )′(t )

a2(t ), ∀t : a(t ) 6= 0;

0, other wi se.

BecauseH ( f )(t )

f (t )= a(t )sin(φ(t ))

a(t )cos(φ(t ))= tan(φ(t ))

and

φ′(t ) = d

d tarctan

(H ( f )(t )

f (t )

)= 1

1+(

H ( f )(t )f (t )

)2

H ( f )(t ) f ′(t )− f (t )H ( f )′(t )

f 2(t )

= f 2(t )

f 2(t )+H ( f )2(t )

H ( f )(t ) f ′(t )− f (t )H ( f )′(t )

f 2(t )= H ( f )(t ) f ′(t )− f (t )H ( f )′(t )

a2(t )(cos2(φ(t ))+ sin2(φ(t ))

= H ( f )(t ) f ′(t )− f (t )H ( f )′(t )

a2(t )

This simply equalsω(t ) =φ′(t ),

where φ(t ) is the phase obtained by the decomposition (1.1) of the signal f .

Chapter 2

Riesz transforms

This chapter will introduce the Riesz transform. The first part of this chapter will state someclassical results on the Riesz transform, give an interpretation for the term “analytical” in thename “analytical signal” and state that the Riesz transforms are steerable.

The second part of this chapter states new results. We will define the Riesz transform of twoclasses of distributions – distributions modulo polynomials (Definition 2.2.10) and tempereddistributions whose support in the Fourier domain does not contain the origin (Theorem 2.2.13).As applications by means of a certain new operator between S ′(R) and S ′(Rn) (Proposi-tion 2.2.14) we state a novel connection between the Hilbert and the Riesz transform (The-orem 2.2.15) and give a novel Bedrosian identity for Hilbert transforms of distributions (Theo-rem 2.2.18) and Riesz transforms of distributions (Theorem 2.2.19).

In the following letRn+1+ denote the upper half space

Rn+1+ := (x, x0) : x ∈Rn , x0 > 0

for which Rn is the boundary hyperplane. Let eαnα=1 ⊂ Rn+1+ be an orthonormal basis of

Rn ⊂Rn+1+ and let e0 ∈Rn+1+ such that eαnα=0 is a orthonormal basis ofRn+1.

2.1 Riesz transforms and conjugate harmonic functions

2.1.1 Riesz transforms

This section will introduce the Riesz transform as the unique extension of the Hilbert trans-form to higher dimensions. I.e. the Riesz transform is the only transform satisfying certainreasonable properties that reduces to the Hilbert transform in the one dimensional case. (SeeTheorem 2.1.2 and Theorem 2.1.7.)

Furthermore we will note the correspondence between the term analytical in the expressionanalytical signal and the term analytical function.

Finally we will state in which respect the Riesz transform is steerable.

Definition 2.1.1 (The partial Riesz transform)LET f ∈ Lp (Rn); 1 < p <∞.

19

20 CHAPTER 2. RIESZ TRANSFORMS

THEN the partial Riesz transforms are defined by

Rα : Lp (Rn ,R) → Lp (Rn ,R), Rα f (x) := limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>δ

yα|y |n+1 f (x − y)d y,

where α ∈ 1, . . . ,n.

The Riesz transform is the operator

R : Lp (Rn ,R) → Lp (Rn ,Rn), f 7→ (R1 f , . . . ,Rn f ).

In the following we will refer to dilations of the form D t f (x) = |t |− n2 f (t x) ∀t > 0, x ∈Rn , as

isotropic dilations.

Theorem 2.1.2 (Properties of the Riesz transformations)LET α ∈ 1, . . . ,n and f ∈ Lp (Rn), where 1 < p <∞.

THEN the following properties of the Riesz transforms hold:

(i) LET ρ ∈ O(n) be a rotation or a reflection which acts on functions byρ( f )(x) := f (ρ−1x).

THEN

ρ−1Rαρ f =n∑β=1

ραβRβ f ;

(ii) Rα commutes with translation;

(iii) Rα commutes with isotropic dilations;

(iv) ∃A(p,n) such that ‖Rα f ‖p ≤ A(p,n)‖ f ‖p , ∀ f ∈ Lp (Rn);

(v) Rα is uniquely defined by properties (i)-(iv) up to a multiplicative complex constant;

(vi) The Fourier multiplier of the Riesz transform is Rα f (x) = i xα|x| f (x);

(vii)∑nα=1 R2

α( f ) =− f ;

(viii) Rα =− ∂∂xα

(−4)−12 .

2.1. RIESZ TRANSFORMS AND CONJUGATE HARMONIC FUNCTIONS 21

Proof. The Proof will follow along the lines of [45] chapter III.

ad (i)

ρ−1Rαρ f (x) = ρ−1Rα f (ρ−1x)

= ρ−1 limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>δ

yα|y |n+1 f (ρ−1(x − y))d y

= limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>δ

yα|y |n+1 f (x −ρ−1 y)d y

= limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>δ

ρyα|ρy |n+1 f (x − y)d y

= limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>δ

∑nβ=1ραβyβ

|y |n+1 f (x − y)d y

=n∑β=1

ραβ limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>δ

yβ|y |n+1 f (x − y)d y

=∑β

ραβRβ f (x).

ad (ii)

Rα(Tl f )(x) = Γ( n+12 )

π(n+1)/2limε→0

∫|y |>ε

yα f (x − l − y)

|y |n+1 d y = Rα( f )(x − l )

= Tl Rα( f )(x).

ad (iii) Let D t : f (x) →|t |− n2 f (t x) denote the dilation operator, then

RαD t f (x) = |t |− n+12 Rα f (t x)

= |t |− n2 limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>δ

yα|y |n+1 f (t (x − y))d y

= |t |− n2 limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>tδ

t−1zα|t−1z|n+1 f (t x − z)t−nd z

= |t |− n2 limδ→0

Γ( n+12 )

π(n+1)/2

∫|y |>tδ

zα|z|n+1 f (t x − z)d z

= D t Rα f (x).

ad (iv) A proof can be found in [45] II.4.2, Theorem 3.

ad (v) First we will proof the following

Lemma 2.1.3LET m :Rn →Rn be homogeneous of degree 0, i.e., m(t x) = m(x), ∀x ∈Rn , t ∈R+,and let

m(ρx) = ρ(m(x)). (2.1)

THEN

∃c ∈R : m(x) = cx

|x| .


Proof of the Lemma. It suffices to proof the Lemma in the case that x ∈ Sn−1. Lete1, . . . ,en be the orthonormal basis of Rn , let c = m1(e1) and let % be any rotation ofRn leaving e1 fixed. Then by (2.1) it follows that

mα(e1) =∑β

%αβmβ(e1)

and hence the (n −1)-dimensional vector (m2(e1), . . . ,mn(e1))T is invariant under all(n −1)-dimensional rotations. Thus m2(e1) = . . . = mn(e1) = 0.

It follows that mα(ρe1) = ρα1m1(e1) = cρα1 for any rotation ρ ofRn . Now let x = ρe1,then ρα1e1 = xα and thus mα(x) = cxα.

Let now Tαnα=1 be a family of operators which satisfy (i)-(iv). Then by (ii) and (iv) it

follows that the Tα can be realized by bounded Fourier multipliers Tα = mα.

Now (iii) shows that

mα(t x) f (x) = t−n2 D t (mα(x) f (

x

t)) = D t (mα(x)D t−1 f (x))

=F (D−t TαD t f )(x) =F (Tα f )(x)

= mα(x) f (x),

i.e. m is homogeneous of degree 0.

Using (i) we see that on the one hand

F (ρ−1Tαρ f )(x) = ρ−1mα(x)(ρ f )(x) = ρ−1(mα(x) f (ρ−1x))

= mα(ρx) f (x)

and on the other hand

F(∑β

ραβTβ f)(x) =∑

β

ραβFTβ f (x) =∑ραβmβ(x) f (x).

Thus the mα suffice (2.1) and thus our Lemma, can be applied. Hence mα = c xα|x| for

some constant c.

ad (vii) To calculate the Fourier multiplier corresponding to the Riesz transform we use the

fact thatΓ( n+1

2 )

π(n+1)/2 xα is a homogeneous harmonic polynomial of degree 1 and Theorem5 in chapter III of [45]:

Lemma 2.1.4LET Pβ(x) be a homogeneous harmonic polynomial of degree β, β≥ 1.

THEN the Fourier multiplier corresponding to the transformation

T f (x) = l i mε→0

∫|y |≥ε

Ω(y)

|y |n f (x − y)d y

withΩ(x) = Pβ(x)

|x|β is γβPβ(x)

|x|β , where γβ = iβπn/2 Γ( β2 )

Γ( β+n2 )

Using this Lemma we get mα = Γ( n+12 )

π(n+1)/2 γ1xα|x| = i xα

|x| .


ad (vi) For the Fourier multipliers (see (vi)) it obviously holds true that

F (n∑α=1

R2α( f ))(t ) =

(−i )(−i )

∑nα=1

t 2α|t | f (t ) =− f (t ) ∀t 6= 0

0 if t = 0

ad (vii) This can easily be seen by computing the corresponding Fourier multipliers.

F(Rα f

)(x) = i

xα|x| f (x) = 2πi xα(−i 24π2|x|2)−1/2 f (x) =F

(− ∂

∂.α(−4)−1/2 f

)(x).

2.1.2 Conjugate harmonic functions

Conjugate harmonic functions are the connection between the term analytical in the expres-sion analytical signal and the term analytical function. Furthermore they add a further respectin which the Riesz transform is the unique extension of the Riesz transform.

In order to define conjugate harmonic functions we will first need to introduce the Poissontransform:

Definition 2.1.5 (The Poisson transform)LET x0 > 0.

THEN the Poisson kernel Px0 onRn is defined by

Px0 (x) :=∫Rn

e2πi ⟨t ,x⟩e−2π|t |x0 d t .

LET f ∈ Lp (Rn), 1 < p <∞.

THEN the Poisson kernel defines via convolution the Poisson transform u f

u f (x, x0) := (Px0 ∗ f )(x).

The Poisson transform maps a function in Lp (Rn) to a function in Lp (Rn+1).

Theorem 2.1.6 (Properties of the Poisson transform)LET f ∈ Lp (Rn), 1 < p <∞, and let u f (x, x0) = (Px0 ∗ f )(x) be its Poisson transform.

THEN

(i) Px0 (x) = Γ( n+12 )

πn+1

2

x0

(|x|2+x20 )

n+12

;

(ii) 4u f = ∂2u f

∂x20+∑n

α=1∂2u f

∂x2α= 0, i.e., the Poisson transform of an Lp -function is harmonic;

(iii) Px0 > 0;

(iv)∫

Rn Px0 (x)d x = 1, ∀x0 > 0;

(v) Px0 (x) is a decreasing function of |x|;(vi) Px0 (x) is harmonic inRn+1+ ;


(vii) Px0 (x) ∈ Lp (Rn), 1 ≤ p ≤∞, and for f ∈ Lp (Rn) it follows that u f ∈ Lp (Rn+1) is harmonicin Rn+1+ ;

(viii) Let u(x, x0) be harmonic in Rn+1+ . Then u is the Poisson transform of a function inLp (Rn) if and only if supx0>0 ‖u(x, x0)‖p <∞;

(ix) Semi-group property: Pl ∗Pr = Pl+r ∀l ,r > 0;

(x) limx0→0 u f (x, x0) = f (x) for almost every x and limx0→0 ‖ f (x)−u(x, x0)‖p = 0, 1 ≤ p <∞;

(xi) Px0 (x) is homogenous of degree −n: Pa(x) = P1(x/a)a−n , a > 0;

(xii) The Fourier multiplier corresponding to the Poisson transform is e−2π|t |x0 ;

Proof. See [45] Chapters III.2 and III.4.3.

The connection between the analytical signal and analytical functions are a set of generalizedCauchy-Riemann equations which the Poisson transforms of the analytical signal respectivelythe Poisson transforms of the Riesz transforms and the original function obey. This theoremcan be found in [45] Theorem 3 in Chapter III.

Theorem 2.1.7 (Conjugate harmonic functions)LET f and fα, α= 1, . . . ,n, belong to Lp (Rn), 1 < p <∞, and let

u0(x, x0) := Px0 ∗ f (x), uα(x, x0) := Px0 ∗ fα(x).

THEN

fα =−Rα( f ), α= 1, . . . ,n,

if and only if the generalized Cauchy-Riemann equations hold:

n∑α=0

∂uα∂xα

= 0, (2.2)

∂uα∂xβ

= ∂uβ∂xα

; α,β= 0, . . . ,n.

Proof. We will give the proof in the case p = 2. The theorem is equivalent to a correspondingtheorem in [45] Chapter III.4.4, a reference for the complete proof can be found there. Notehowever that the Fourier transform is defined differently there.

Suppose fα =−Rα f , then fα(t ) = −i tα|t | f (t ) and, hence,

uα(x, x0) =∫Rn

f (t )−i tα|t | e−2π|t |x0 e2πi t x d t ,

and

u(x, x0) =∫Rn

f (t )e−2π|t |x0 e2πi t x d t .


Because of the dominated convergence theorem [41], we may differentiate under the integralsign, and hence,

∂u

∂x0(x, x0) =−2π

∫Rn

f (t )|t |e−2π|t |x0 e2πi t x d t .

∂u

∂xβ(x, x0) = 2πi

∫Rn

f (t )tβe−2π|t |x0 e2πi t x d t .

∂uβ∂x0

(x, x0) = 2πi∫Rn

f (t )tβe−2π|t |x0 e2πi t x d t .

∂uα∂xβ

(x, x0) = 2π∫Rn

f (t )tαtβ|t | e−2π|t |x0 e2πi t x d t .

Now the generalized Cauchy-Riemann conditions are easy to check. The first one follows from

n∑α=1

∂uα∂xα

(x, x0) =n∑α=1

2π∫Rn

f (t )t 2α

|t |e−2π|t |x0 e2πi t x d t

= 2π∫Rn

f (t )

∑nα=1 t 2

α

|t | e−2π|t |x0 e2πi t x d t

= 2π∫Rn

f (t )|t |e−2π|t |x0 e2πi t x d t

=− ∂u

∂x0(x, x0),

the second equation follows from

∂uα∂xβ

(x, x0) = 2π∫Rn

f (t )tαtβ|t | e−2π|t |x0 e2πi t x d t

= ∂uβ∂xα

(x, x0),

and

∂u

∂xβ(x, x0) = 2πi

∫Rn

f (t )tβe−2π|t |x0 e2πi t x d t

= ∂uβ∂x0

(x, x0).

Conversely, let α ∈ 1, . . . ,n and uα(x, x0) = ∫Rn fα(t )e−2π|t |x0 e2πi t x d t . The fact that ∂u0

∂xα= ∂uα

∂x0,

shows that

2πi∫Rn

f (t )tαe−2π|t |x0 e2πi t x d t =−2π∫Rn

fα(t )|t |e−2π|t |x0 e2πi t x d t .

Therefore by the uniqueness of the inverse Fourier transform fα(t ) =− i tα|t | f (t ), and so

fα =−Rα( f ), α= 1, . . . ,n.


Remark 2.1.8The equations (2.2) are not fully satisfactory because the analytical signal satisfying this equa-

tion would be

(f

−H f

)– that is a vector valued function rather than fa = f + iH f the com-

plex valued monogenic signal.

The reason for this difference is that equations (2.2) do not reduce to the Cauchy-Riemannequations of complex analysis.

To fix this in Theorem 4.5.2 we will define the analog of the analytical signal for Riesz trans-forms – the monogenic signal in chapter 4 and the analog of complex analysis for higher di-mensions – Clifford analysis in section 3.3.

2.1.3 Steerabilty of the Riesz transform

R(1,0) = R1 R(2−1/2,2−1/2) = 2−1/2(R1 + R2)

R(cos(3π/16),sin(3π/16)) R(0,1) = R2

Figure 2.1: Steerability of the Fourier multiplier of the Riesz transform: In the upper left and inthe lower right corner the Fourier multipliers of the Riesz transforms with respect to the basisdirections are given, the other two images portrait linear combinations - the Fourier multipli-ers with respect to the directions corresponding to the linear combination of the basis direc-tions.

LET E = e1, . . . ,en and D = d1, . . . ,dn be two ONBs of Rn . A linear orthogonal mappingρ ∈O(Rn) is then uniquely defined by dl = ρel , ∀l = 1, . . . ,n.

THEN according to Theorem 2.1.2[i] the Riesz transforms with respect to the basis D can be

2.2. RIESZ TRANSFORMS OF DISTRIBUTIONS 27

expressed via the Riesz transforms with respect to the basis E as

Rdlf (x) =

n∑β=1

ρl ,βReβ (x).

As a consequence the Riesz transform with respect to an arbitrary direction is well defined asa linear combination of the Riesz transforms with respect to the basis directions.

Definition 2.1.9 (Steerability of the Riesz transform)LET d ∈Rn : |d | = 1 and let ρ ∈ SO(Rn) : d = ρe1.

THEN the Riesz transform in direction d is given by

Rd f (x) = ρ−1R1ρ f (x) =n∑β=1

ρ1,βRβ f (x), ∀ f ∈ L2(Rn), x ∈Rn . (2.3)

From this definition it is clear, that the Riesz transform is steerable, since the Riesz transformwith respect to any direction is a linear combination of the n Riesz transforms with respect tothe basis directions. This is illustrated in Figure 2.1.

2.2 Riesz transforms of distributions

In this section we will define the Riesz transform of a distribution. To do this we will considerthe Lizorkin space, a subspace of the Schwartz space which is invariant under the Riesz trans-form. Then we will define the Riesz transform for elements in the dual of this space, which isthe space of tempered distributions modulo polynomials and for a certain class of tempereddistributions.

The Riesz transform on the dual of the Lizorkin space we define seems to be new as well asthe Riesz transform on the subset of tempered distributions we will consider. Since the tem-pered distributions we consider are not necessarily compactly supported they complementthe known theory of Riesz transforms of compactly supported distributions. The Hilbert trans-form of a compactly supported distribution is easily defined via convolution with the Cauchy

principal value distribution PV∫R

φ(x)x d x, ∀φ ∈ D (Rn). (See [30] for details on the Hilbert

transforms of distributions.)

In subsection 2.2.6 it will become apparent that our definition of the Riesz transform of distri-butions is especially well suited to proof a novel Bedrosian identity for distributions. Indeedthe prerequisites for the Bedrosian identity are all that is required for the Hilbert transformof the involved distributions to be well defined. Furthermore we will give the connection be-tween the Riesz transform of certain distributions and the Hilbert transform. This connectionwill then yield a Bedrosian identity for Riesz transforms of certain distributions.

2.2.1 Preliminaries: Lizorkin spaces and derivatives of the Riesz multiplier

Definition 2.2.1 (Lizorkin spaces)The Lizorkin spaceΦ(Rn) is the linear subspace of S (Rn) defined as

Φ(Rn) =φ ∈S (Rn) :

∫Rn

xγφ(x)d x = 0, ∀γ= γ1, . . . ,γn ∈Nn0

.


Thus the Lizorkin space is the subspace of the Schwartz space that consists of functions witharbitrarily many vanishing moments.

Proposition 2.2.2 (Lizorkin spaces and distributions)LET Φ(Rn) be the Lizorkin space.

THEN Φ(Rn) is a closed subspace of S (Rn).

LET Ψ(Rn) = ψ ∈S (Rn) : ∂γψ(0) = 0, ∀γ= γ1, . . . ,γn ∈Nn

0

.

THEN Φ(Rn) and Ψ(Rn) are a Fourier pair, i.e., F :Φ(Rn) →Ψ(Rn) and F :Ψ(Rn) →Φ(Rn).The topologies on Φ(Rn) ⊂ S (Rn) and Ψ(Rn) ⊂ S (Rn) are the subspace topologies. Thetopological dualΦ′(Rn) is the space of tempered distributions modulo polynomials.

Proof. See [31].

Remark 2.2.3The Lizorkin spaceΦ(Rn) is often denoted as S0(Rn) [25] or even S∞(Rn).

Definition 2.2.4 (The space E0(Rn))Let us denote E0(Rn) := f ∈ C∞(Rn) : ∂γ f (0) = 0, ∀γ ∈ Nn

0 .ObviouslyΨ(Rn) =S (Rn)∩E0(Rn).

We will use the following characterization of E0 from [31].

Lemma 2.2.5 (Characterizations of E0)LET f ∈C∞(Rn).

THEN the following are equivalent:

1. f ∈ E0(Rn).

2. lim|x|→0

∂γ f (x)

|x|l = 0; ∀γ ∈Nn0 , l ∈N0.

3. lim|x|→0

f (x)

|x|l = 0; ∀l ∈N0.

The above Lemma implies that to proof that the Riesz transform maps the Lizorkin spaceΦ(Rn) into itself, we have to take a look at the partial derivatives of the Fourier multiplier

Rα :Rn →R, ξ 7→ iξα|ξ|

of the partial Riesz transform.

2.2.2 Derivatives of the Fourier multiplier of the Riesz transform

Lemma 2.2.6 (Partial derivatives of Rα)LET β= (β1, . . . ,βn) ∈Nn

0 be a multi index, α ∈ 1, . . . ,n, ξ ∈Rn \ 0.

THEN

∂βξα

|ξ| =βαΘ(ξ;β1, . . . ,βα−1,βα−1,βα+1, . . . ,βn)+ βα!

2(βα−2)!ξαΘ(ξ,β1, . . . ,βn). (2.4)


Here,

Θ(ξ;β1, . . . ,βn) =(−1)β

∑nj=1

1β j !

|ξ|2|β|+1ξβ

+ ∑k1(β1),k2(β1)∈Tβ1

...k1(βn ),k2(βn )∈Tβn

β!

k1(β)!k2(β)!

n∑j=1

1(k1(β j )+k2(β j )

)!

(−1)k1(β)(−1)k2(β)

|ξ|2(k1(β)+k2(β))+1ξk1(β)

As usual β! =∏nj=1β j ! and a|β| = a

∑nj=1β j ; ∀a ∈R.

Tβ j denotes the set of pairs of integers (k1,k2) ∈N20 such that β j = k1(β j )+2k2(β j ) and kl (β) =(

kl (β1), . . . ,kl (βn)).

Proof. For α ∈ 1, . . . ,n and ξ ∈Rn \ 0 we can write

ξα

|ξ| =ξα√∑nj=1 ξ

2j

= f (ξα )g (h(ξ)),

where f (a) = a, g (a) = a− 12 ∀a ∈R and h(ξ) =∑n

k=1 ξ2k , ∀ξ ∈Rn . Using the product rule on the

α-th partial derivative we get

∂βα

∂ξβαα

f (ξα)g (h(ξ)) =βα∑

k=0

(βα

k

)∂βα−k

∂ξβα−kα

f (ξ j )∂k

∂ξkα

g (h(ξ))

=βα ∂βα−1

∂ξβα−1α

g (h(ξ))+ βα!

2(βα−2)!ξα

∂βα

∂ξβαα

g (h(ξ)).

Now we apply the remaining partial derivatives and set

Θ(ξ;β1, . . . ,βn) = ∂∑n

k=1βk g (h(ξ))∏nk=0∂ξ

βkk

.

The formula of Faá di Bruno Lemma A.1.4 which yields (2.4).

Since in our case h(ξ) = c +ξ2, where c ∈R is independent of ξ, only the subset of Tm , whereonly coefficients b1 and b2 are non-zero, is relevant.

Now the statement follows since g (m)(ξ) = cξ−12 −m and h′(ξ) = 2ξ and hence h(2)(ξ) = 2. The

constants in the statement follow since the constant factors 2 in the derivatives of h cancel thefactors of 1

2 in the derivatives of g .

Remark 2.2.7From the above Lemma it follows that the partial derivatives of Rα have the form

∂βξα

|ξ| =pα,β(ξ)

q2|β|+1(|ξ|) , ∀ξ ∈Rn \ 0, (2.5)

where

pα,β(ξ) =β1∑

k1=0· · ·

βα+1∑kα=0

· · ·βn∑

kn=0akξ

k ,

and ak ∈C, k = (k1, . . . ,kn) and q2|β|+1(|ξ|) =∑2|β|+1j=0 b j |ξ| j , and b j ∈C.


Theorem 2.2.8 (Partial derivatives of Rα are homogenous)The partial derivatives of Rα have the form

∂βξα

|ξ| =hα,β(ξ)

|ξ||β| , ∀ξ ∈Rn \ 0, (2.6)

where hα,β :Rn →R is a 0-homogenous function.

Proof. Note that ∀ε > 0, ξα|ξ| ∈ C∞(

Rn \ Bε(0)), whence multiindex partial derivatives are well

defined. Let βα := (β1, . . . ,βα−1,βα + 1,βα+1, . . . ,βn). We will use induction over |β| to proofthat for all ξ ∈Rn \ 0 the derivatives are a sum of terms of the form

cα,γ,βξγ

|ξ||γ|+|β| , (2.7)

where γ≤βα and cα,γ,β ∈C.

This is clear for |β| = 0, since then βα = (δα,k )nk=1 which yields ξα

|ξ| .

Let |β| = k, ν ∈ 1, . . . ,n and assume that ∂β ξα|ξ| consists of terms of the form (2.7).

Then |βν| = k+1 and ∂βν ξα|ξ| = ∂

∂ξν∂β

ξα|ξ| . Hence we only need to proof that any partial derivative

of degree one of a term of the form (2.7) is again a sum of terms of this form.

Let γ≤βα. By the product rule

∂

∂ξν

ξγ

|ξ||γ|+|β| =ξ(γ1,...,γν−1,γν+1,γν+1,...,γn )

|ξ||γ|+2+|β| + (1−δγν,0)γνξ(γ1,...,γν−1,γν−1,γν+1,...,γn )

|ξ||γ|+|β|

Notice that for any multiindex µ ∈Nn0 the functionRn →R, ξ 7→ ξµ

|ξ||µ| is 0-homogenous. Now

in the first term the degree of the monomial is increased by one, whereas the power in thedenominator is increased by two. In the second term, which only exists if γν 6= 0, the degree ofthe monomial in the numerator is decreased by one, whereas the power in the denominator

remains unchanged. As a consequence both terms are of the form ξµ

|ξ||µ|+|βν | , where µ≤βν.

2.2.3 Riesz transforms of distributions modulo polynomials

Theorem 2.2.9 (Riesz transform and the Lizorkin space)LET α ∈ 1, . . . ,n.

THEN the partial Riesz transform Rα :Φ(Rn) →Φ(Rn) is a bounded linear mapping.

Proof. Let φ ∈Φ(Rn), which is equivalent to

φ ∈Ψ(Rn) = ψ ∈S (Rn) : ∂γψ(0) = 0 ∀γ= γ1, . . . ,γn ∈Nn

0

.

Remember that the partial Riesz transform Rα corresponds to a multiplication by the Fourier

multiplier Rα = iξα|ξ| in the Fourier domain.

To show thatRαφ ∈Φ(Rn)

we will show that Rαφ ∈Ψ(Rn).


We start by proving Rαφ ∈ C∞(Rn), ∀φ ∈ Φ(Rn). Since ∀ε > 0 the Fourier multiplier Rα is anelement of C∞(Rn \ Bε(0)) it is clear that Rαφ ∈C∞(Rn \ Bε(0)), ∀ε> 0.

Furthermore Rα = iξα|ξ| ∈C (Rn \ 0), whence Rαφ ∈C (Rn \ 0).

For ξ = 0 we set Rαφ(0) = 0. Since the Fourier multiplier of the Riesz transform is iξα|ξ| and

since ξαφ(ξ) ∈Ψ(Rn) by Lemma 2.2.5 it follows that limξ→0 Rαφ(ξ) = limξ→0ξαφ(ξ)|ξ| = 0. HenceRαφ ∈C (Rn).

Let γ= (γ1, . . . ,γn) ∈Nn0 , using Remark 2.2.7 and letting p, q as in Lemma 2.2.5

limξ→0

Rαφ(γ)

(ξ) = limξ→0

γ1∑k1=0

. . .γn∑

kn=0

n∏l=1

(γl

kl

)Rα

(γ−k)(ξ)φ(k)(ξ)

= limξ→0

γ1∑k1=0

. . .γn∑

kn=0

n∏l=1

(γl

kl

)pα,γ−k (ξ)φ(k)(ξ)

q2|γ−k|+1(ξ)

= 0,

where k = (k1, . . . ,kn) ∈Nn0 .

To show continuity of Rαφ(γ)

on Rn it remains to show Rαφ(γ)

(0) = 0, ∀γ ∈Nn0 . This can be

done inductively. We have already shown this forγ= 0. Assume Rαφ(γ)

(0) = 0 has already been

proven and let γl+ be the multiindex such that γl+j :=

γ j +1 if j = l

γ j if j 6= l ..

Then Rαφ(γl+)

(0) = limh→0

(Rαφ(γ)

(0)−Rαφ(γ)

(hel ))/h = 0.

Finally to proof Rαφ ∈ C∞(Rn) we show that Rαφ(γ)

(ξ) is bounded on Rn . Let ε > 0. ThenRαφ(γ)

(ξ) is clearly bounded for ξ ∈Rn \(Bε(0)). Since we have shown that Rαφ(γ)

is continuouson the compact set Bε(0) we know that it is bounded on this set, too.

Thus we have shown Rαφ ∈C∞(Rn).

Next we will show that Rαφ ∈S (Rn). Let γ ∈Nn0 and β ∈N0.

supξ∈Rn (1+|ξ|)β|∂γ Rα(φ)(ξ)| = supξ∈Rn

(1+|ξ|)β∣∣∂γ(iξα

|ξ| φ)(ξ)

∣∣= supξ∈Rn

(1+|ξ|)β∣∣∣∣∣ γ1∑k1=0

· · ·γn∑

kn=0

n∏j=1

(γ j

k j

)∂γ−k ξα

|ξ|∂kj φ(ξ)

∣∣∣∣∣= supξ∈Rn

(1+|ξ|)β∣∣∣∣∣∣γ1∑

k1=0· · ·

γn∑kn=0

n∏j=1

(γ j

k j

)pα,γ−k (ξ)

∂k j

j φ(ξ)

q2|γ|+1(|ξ|)

∣∣∣∣∣∣ , (2.8)

where pα,γ and q2|γ−k|+1 as in Remark 2.2.7.

For |ξ| > 1 (2.8) is clearly bounded since φ ∈S (Rn) andpα,γ

q2|β|+1is bounded. Now (2.8) is contin-

uous on the compact set ξ ∈Rn : |ξ| ≤ 1 and hence it is bounded.

This shows that Rαφ ∈Ψ(Rn) and thus that

Rαφ ∈Φ(Rn), ∀φ ∈Φ(Rn).


Definition 2.2.10 (Riesz transform of a distribution inΦ′)LET f ∈Φ′(Rn), α ∈ 1, . . . ,n.

THEN the Riesz transform Rα :Φ′(Rn) →Φ′(Rn) is defined by

Rα f (φ) := f (−Rαφ). (2.9)

Corollary 2.2.11 (The Riesz transform of a distribution inΦ′ is well defined)LET α ∈ 1, . . . ,n.

THEN the Riesz transform Rα :Φ′(Rn) →Φ′(Rn) as defined in Definition 2.2.10 is well defined.

Proof. We will first show that the definition is coherent with our definition of the Riesz trans-form on Lp (Rn), 1 < p <∞. Let 1 < p <∞, f ∈ Lp (Rn), and let φ ∈Φ(Rn).

Rα f (φ) =∫Rn

Rα f (x)φ(x)d x

=∫Rn

ixα|x| f (x)F−1(φ)(x)d x

=∫Rn

f (x)ixα|x| φ(−x)d x

=∫Rn

f (−x)−i xα|x| φ(x)d x

=∫Rn

F−1( f )(x) −Rαφ(x)d x

= f (−Rαφ).

Secondly we need to show that Rα(Φ(Rn)) ⊆ Φ(Rn). This was shown in Theorem 2.2.9 andhence ∀φ ∈Φ : Rαφ ∈Φ(Rn).

Finally we have to proof that Rα f is continuous. Let φk k∈N ⊂Φ(Rn) such that limk→∞φk = 0.In order to show that Rα f is continuous we have to show that limk→∞ Rα f (φk ) = 0. But sinceRα is linear and bounded and hence continuous onΦ(Rn) it follows that limk→∞ Rαφk = 0 andhence limk→∞ Rα f (φk ) =− limk→∞ f (Rαφ) = 0 since f is continuous.

Altogether it follows that R f ∈Φ′.

2.2.4 Riesz transforms of certain tempered distributions

While the Riesz transform is not well defined for all tempered distributions it is well definedfor those whose support does not contain 0.

Definition 2.2.12 (Riesz transform of tempered distributions)LET f ∈S ′(Rn) such that 0 ∉ supp f .

Let τ ∈C∞(Rn) such that0 ∉ supp(τ), τ(x) = 1, ∀x ∈ supp( f ). (2.10)

THEN the Riesz transform

Rα :

f ∈S ′(Rn) : 0 ∉ supp( f )→

f ∈S ′(Rn) : 0 ∉ supp( f )

is defined byRα f (φ) = f

(τF−1(−Rαφ)

), ∀φ ∈

f ∈S (Rn) : 0 ∉ supp( f ). (2.11)


Theorem 2.2.13 (Riesz transform of a distribution in S ′)LET f ∈S ′(Rn) such that 0 ∉ supp( f ) and τ ∈C∞(Rn) such that (2.10) holds.

THEN the Riesz transform defined by

Rα f (φ) = f(τF−1(−Rαφ)

), ∀φ ∈S (Rn) (2.12)

is well defined and independend on the choice of τ.

Proof. Since the support of f is a closed set that does not contain 0 there is a closed set con-taining 0 in the complement of supp( f ). Hence it is indeed possible to choose τ as statedabove.

We will proof that the Riesz transform is independent on the choice of τ. Let σ ∈ C∞(Rn)be another function such that σ(x) = 1, ∀x ∈ supp( f ), 0 ∩ supp(σ) = ;. Furthermore letφ ∈S (Rn). Then

f(τF−1(−Rαφ)

)− f(σF−1(−Rαφ)

)= f

((τ−σ)F−1(−Rαφ)

)∗= 0.

To see that ∗ holds note that (τ−σ)F−1(φ) ∈Ψ(Rn) ⊂S (Rn) and thus by Theorem 2.2.9 that(τ−σ)F−1(−Rαφ) ∈Ψ(Rn) ⊂S (Rn). ∗ now follows since supp((τ−σ)(F−1(φ))∩supp( f ) =;.

The proof that this Riesz transformation is coherent with the Riesz transform on Lp (Rn),1 < p <∞, is analogous to the proof of this fact in Definition 2.2.10.

2.2.5 Hilbert and Riesz transforms

The next theorem states a connection between the Hilbert transform of distributions in S ′(R)and the Riesz transforms of certain distributions in S ′(Rn) which are derived from distribu-tions in S ′(R) using the operator Ld defined below. This operator maps a one dimensionalregular distribution g ∈ S ′(R) to a n dimensional regular distribution Ld (g ) = g (⟨ · ,d⟩) ∈S ′(Rn). (See Example 2.2.1) Ld (g ) is sometimes called a plane wave.

Proposition 2.2.14 (The operator Ld )LET f , g ∈S ′(R), d ∈ Sn−1 and let ρd ∈ SO(n) : ρd (d) = en .

Let Ld : S ′(R) →S ′(Rn) be defined by

Ld f (φ) =∫Rn−1

f (ρ(φ)(x, · ))d x, ∀φ ∈S (Rn).

THEN àLd ( f )(φ) = f (ρφ(0Rn−1 , · )).

Furthermore

Ld ( f )+Ld (g ) =Ld ( f + g ).

If additionally supp(g ) is bounded

Ld ( f )Ld (g ) =Ld ( f g ).


If supp( f ) ⊆R\ Bε(0) for some ε< 0 then

supp( Ld f ) ⊆ x ∈Rn : |⟨x,d⟩| ≥ ε. (2.13)

If supp( f ) ⊆ Bε(0) for some ε> 0 then

supp( Ld f ) ⊆ x ∈Rn : |⟨x,d⟩| ≤ ε.

Proof. We have to show that Ld is well defined, that is independent on the choice of ρd andthat it maps into S ′(Rn).

To show the last it suffices to show that L ( f ) is a direct product of distributions. Then Theo-rem A.2.7 states the result. Let us define for a rotation ρ ∈ SO(n) and a distribution g ∈S ′(Rn)the operator ρ : S ′(Rn) → S ′(Rn) via ρ(g )(φ) := g (ρφ). The operator ρ is obviously welldefined, since the operator ρ : S (Rn) → S (Rn) introduced in the preliminaries in Defini-tion 1.3.21 maps test functions to test functions.

Let I (φ) := ∫Rn−1 φ(x)d x, ∀φ ∈ S (Rn−1). Then Ld = ρd (I × f ) and hence well defined as a

direct product of distributions.

We will now show that our definition is independent of the choice of ρd . Let ρ,σ ∈ SO(n) betwo rotations such that ρ(d) = σ(d) = en . Than there exists a rotation τ ∈ SO(n −1), such that

ρ =σ(τ 00 1

). Hence∫

Rn−1f (σ(φ)(x, · ))d x =

∫Rn−1

f (σφ(τy, · ))d y

=∫Rn−1

f (σ

(τ 00 1

)φ(y, · ))d y

=∫Rn−1

f (ρφ(y, · ))d y

=Ld ( f )(φ), ∀φ ∈S (Rn).

Using Theorem 1.3.12 and Theorem 1.3.22àLd ( f )(φ) = ρ(I × f )(φ) = I × f (ρφ) = I × f (ρφ)

= f (ρφ(0Rn−1 , · )).

Let g ∈S ′(Rn). Then

(Ld ( f )+Ld (g )

)(φ) =

∫Rn−1

f (ρdφ(x, · ))d x +∫Rn−1

g (ρdφ(x, · ))d x

=∫Rn−1

( f + g )(ρdφ(x, · ))d x

=Ld ( f + g )(φ).

Let furthermore supp(g ) be bounded. By Corollary A.2.13 f g is well defined, by Theorem A.2.11g ∈C∞(Rn) and by Theorem A.2.10 f g (φ) = f (gφ), ∀φ ∈S (Rn). Hence(

Ld (g )Ld ( f ))(φ) =

∫Rn−1

f(ρd

(g (⟨(x, · ),d⟩)φ(x, · )

))d x

=∫Rn−1

f(g ( · )ρd

(φ(x, · )

))d x

=Ld (g f )(φ), ∀φ ∈S (Rn).


To proof (2.13) note that

x ∈ supp(àLd ( f )) ⇔ (ρ−1x

)n ∈ Bε(0) ⇔|⟨x,d⟩| ∈ Bε(0)

The following example will illustrate the effect of the operator Ld .

EXAMPLE 2.2.1:Let f ∈C∞(R). Then f defines a distribution in S ′(R) via

f (φ) =∫R

f (t )φ(t )d t , ∀φ ∈S (R).

It follows that

Len ( f )(φ) =∫Rn−1

∫R

f (xn)φ(x1, . . . , xn−1, xn)d x1 . . .d xn−1d xn , ∀φ ∈S (Rn).

And

Ld ( f )(φ) =∫Rn−1

∫R

f (xn)ρdφ(x1, . . . , xn−1, xn)d x1 . . .d xn−1d xn =∫Rn

f ((ρ−1 y)n)φ(y)d y

=∫Rn

f (⟨y,d⟩)φ(y)d y, ∀φ ∈S (Rn).

That is, in distributional sense Ld ( f ) = f (⟨ · ,d⟩).

Theorem 2.2.15 (Hilbert and Riesz transforms)LET f ∈S ′(R) such that 0 ∉ supp( f ). Furthermore let d ∈Rn : ‖d‖ = 1.

THEN the Riesz transform of Ld ( f ) ∈S ′(Rn) exists. Indeed

Rd Ld ( f ) =Ld (H f ),

andRl Ld ( f ) = 0, ∀l ∈Rn : ⟨d , l⟩ = 0.

Proof. Since R \ 0 is an open set, while supp( f ) is closed, ∃ε> 0 : Bε(0)∩ supp( f ) =;. It fol-lows that 0 ∈ x ∈Rn : |⟨x,d⟩| < ε and by (2.13) it follows that

x ∈Rn : |⟨x,d⟩| < ε∩ supp(àLd ( f )) =;.

Let us choose τ ∈ C∞(R) satisfying (2.10) for f ∈ S ′(R), that is supp(τ) ∩ 0 = ; andτ(x) = 1, ∀x ∉ Bε(0). Then τd (x) = τ(⟨x,d⟩) satisfies (2.10) and we can apply Definition 2.2.12to Ld ( f ).

Rd Ld ( f )(φ) = àLd ( f )(−Rdτd F−1(φ)

)= f

(ρ(− Rdτd F−1(φ)

)(0Rn−1 , · )

)= f

(i − ·|(0Rn−1 , · )|τ

(⟨(0Rn−1 , · ),en⟩)ρ(F−1(φ))(0Rn−1 , · )

)= f

(i sgn( · )τ( · )ρ

(φ(0Rn−1 ,− · )

))=Ld (H f )(φ), ∀φ ∈S (Rn).


Furthermore, for any l ∈ Sn−1 such that ⟨l ,d⟩ = 0 there exists some al ∈ Rn−1 withρd (l ) = (al ,0). Hence

Rl Ld ( f )(φ) = àLd ( f )(− Rlτd F−1(φ)

)= f

(ρ(− Rlτdφ

)(0Rn−1 , · )

)= f

(i 0R|(0Rn−1 , · )|τ

(⟨(0Rn−1 , · ),en⟩)ρ(φ)(0Rn−1 , · )

)= f

(0C∞(Rn )ρ(φ)(0Rn−1 , · )

)= 0S ′(Rn )(φ), ∀φ ∈S (Rn).

2.2.6 The Bedrosian identity and Riesz transforms

The Bedrosian identity has first been proven by Bedrosian in [5]. The purpose is to determineunder which conditions the amplitude phase decomposition achieved by the analytical signalgives the phase and amplitude one would expect by the construction of the signal. That is, leta ∈ L2(R), φ :R→R : cos(φ) ∈ L2(R) and let f = a cos(φ) ∈ L2(R). The Bedrosian identitystates conditions on a and cos(φ) under which fa = a exp(φ).

In this section we will extend the Bedrosian identities to distributions and give a Bedrosianidentity for Riesz transforms.

Let us however first state the classical Bedrosian identity [5].

Theorem 2.2.16 (The Bedrosian identity)LET f , g ∈ L2(R) such that supp( f ) ⊆ [a,b] and supp(g ) ⊂R\ [−b,−a].

THEN

H ( f g ) = f H (g ). (2.14)

Remark 2.2.17The original Bedrosian identity is given for functions in f , g ∈ L2(R). It is at first view not clearif H f g ∈ L2(R) or in which sense this identity should be interpreted, it is simply assumed tobe well defined in some sense.

Indeed since supp( f ) is bounded, f ∈ L1(R), and by Corollary A.2.12 it holds that f ∈C∞(R) isbounded and thus f g ∈ L2(Rn).

In the case of a Bedrosian identity for distributions we have to be more careful: It does holdtrue that f g ∈ S ′(R) for f ∈ S ′(Rn) with compact support in the Fourier domain and g ∈S ′(R). But f g ∈ S ′ is not enough to ensure that the Riesz transform is well defined. It isremarkable that this well definedness is guaranteed by the same condition on the support ofthe distributions that is needed for the Bedrosian identity to hold.

Theorem 2.2.18 (The Bedrosian identity for distributions)LET a < 0 < b and let f ∈ S ′(R) such that supp( f ) ⊆ [a,b] and g ∈ S ′(R) such thatsupp(g ) ⊂R\ [−b,−a].

THEN

H ( f g ) = f H (g ). (2.15)


Proof. Since the support of a distribution is a closed set,

∃ε> 0 : supp(g ) ⊆R\]−b −ε,−a +ε[.

Let φ ∈S (R).

First we have to show that H f g and f H g are well defined in S ′(R).

Since supp( f ) is bounded the convolution f ∗ g is well defined by Theorem A.2.14 as a directproduct. Furthermore 0 ∉ supp( f g ):

f g (φ) = f ∗ g (φ) = f × g(φ( · f + · g )

), ∀φ ∈S (Rn)

If x + y ∈]− ε,ε[ then either x ∉ [a,b] = supp( f ) or y ∉ R\]− b − ε,−a + ε[= supp(g ). Hence0 ∉ supp( f g ).

Hence f g satisfies the condition of Theorem 2.2.13, which states that H f g ∈S ′(R).

Now g satisfies the conditions of Theorem 2.2.13 as well, that is H g ∈ S ′(R) and f H g ∈S ′(R) by Theorem A.2.14.

Now let σ f ,τg ,µ f g ∈C∞(R) such that

σ f (x) = 1 ∀x ∈ [a,b], supp(σ f ) ⊆ [a −ε/2,b +ε/2],

τg (y) = 1, ∀y ∉]−b −ε,−a +ε[, supp(τg )∩ [−b −ε/2,−a +ε/2] =;,

µ f g (z) = 1, ∀z ∉ [−ε,ε], supp(µ f g )∩ [−ε/2,ε/2] =;.

It follows that κx (y) := µ f g (x + y)σ f (x) ∈ C∞(R) and κx (y) = 1, ∀y ∈ [−ε−b,−a +ε]. Conse-quently σ f satisfies (2.10) for f , τgκx satisfies (2.10) for g and µ f g satisfies (2.10) for f g .

H f g (φ)2.2.12= f g

(−H µ f g φ∨) A.2.14= f ∗ g

(−H µ f g φ∨)

(2.16)

(1)= f(g(− i sgn( · f + · g )µ f g ( · f + · g )φ(− · f − · g )

))(2)= f

(g(− i sgn( · f + · g )σ f ( · f )τg ( · g )µ f g ( · f + · g )φ(− · f − · g )

))(3)= f

(g(− i sgn( · g )σ f ( · f )µ f g ( · f + · g )τg ( · g )φ(− · f − · g )

))= f

(g(− i sgn( · g )κ · f ( · g )τg ( · g )φ(− · f − · g )

))2.2.12= f

(H g(φ(− · f − · g )

))(1)= f ∗H g (φ∨)

= f H g (φ),

where φ∨(x) =φ(−x), ∀x ∈R.

(1) is the convolution of the two distributions f and H g . This convolution is well defined sincef has bounded support. (See Theorem A.2.14.)

(2) We used the fact that if h ∈S ′(R), ν ∈C∞(R) : ν(x) = 1, ∀x ∈ supp(h) and φ ∈S ′(R) then

h(φ) = h(νφ+ (1−ν)φ) = h(νφ).


(3) holds true since sgn(x + y) = sgn(y), whenever

x + y > 0∧ y > 0

orx + y < 0∧ y < 0.

Now this holds true if x ∈ supp(σ f ) ⊆ [a,b] and y ∈ supp(τg ) ⊆R\]−b −ε, a +ε[.

EXAMPLE 2.2.2:The calculation (2.16) is made somewhat intransparent by the necessary use of the functionsσ,τ,µ and κ. Therefore we will now calculate the same fact under the additional assumptionthat f , g ∈ L1

loc. Then the integration bounds will take over the role of the functions σ,τ,µ andκ.

H f g (φ)2.2.12= f g

(−H µ f g φ∨) A.2.14= f ∗ g

(−H µ f g φ∨)

(1)=∫

[a,b]f (x)

∫R\[−ε−a,ε−b]

−i sgn(x + y)g (y)µ f g (x + y)︸︷︷︸=1

φ(−x − y)d xd y

(2)=∫

[a,b]f (x)

∫R\]−ε−b,ε−a[

−i sgn(y)g (y)φ(−x − y)d xd y

2.2.12= f(H g

(φ(− · f − · g )

))(1)= f ∗H g (φ∨)

= f H g (φ)

(1) is the convolution of the two distributions f and H g . This convolution is well defined sincef has bounded support. (See Theorem A.2.14.)

(2) holds true since sgn(x+ y) = sgn(y), whenever x+ y > 0∧ y > 0 or x+ y < 0∧ y < 0. Now thisholds true since supp( f ) ⊆ [a,b] and supp(g ) ⊆R\]−b −ε, a +ε[.

Theorem 2.2.19 (A Bedrosian identity for the Riesz transform)LET f , g ∈S ′(R) such that supp( f ) ⊆ [−R,R] and supp(g ) ∈R \ [−R,R] for some R ∈R+. Fur-thermore let d ∈ Sn−1.

THEN F =Ld f and G =Ld g are in S ′(Rn) and satisfy the Bedrosian identity

R(FG) = F R(G).

Proof. We will first show that F ∈ C∞(Rn). By Theorem A.2.11 f ∈ C∞(R) since the supportof its Fourier transform is bounded. Furthermore | f (x)| ≤ γ(1 + |x|)N , ∀x ∈ R, where N isthe order of the distribution f which is finite. Since derivatives in the time respectively spacedomain correspond to multiplication by a polynomial on the Fourier domain Theorem 1.3.17,we can once again use Theorem A.2.11 to get

∀p ∈N∃N ∈N,γp,N > 0 : |Dp f (x)| ≤ γp,N (1+|x|)N , ∀x ∈R. (2.17)

If f ∈C∞(R), then F = f (⟨ · ,d⟩) ∈C∞(Rn). By (2.17) and since d ∈ Sn−1 it follows that

∀α ∈Nn0 |DαF (x)| ≤ γp,N (1+|x|)Np , ∀x ∈Rn ,


where p = |α| =∑nl=1αl . Furthermore, by construction G ∈S ′(Rn). Thus by Theorem A.2.10(i)

it follows that FG ∈S ′(Rn) is well defined.

To proof that the Riesz transform of FG is well defined in S ′(Rn) it is sufficient to show that0 ∉ supp(FG).

Since FG ∈S ′(Rn) by Theorem A.2.14 we know that FG = F ∗G . That is, we can proceed as inthe proof of Theorem 2.2.18. We recall that, since the support of a distribution is a closed set∃ε> 0 : supp(g ) ⊆R\]−R −ε,R +ε[. It follows that

FG(φ) = F ∗G(φ) = F(G

(φ( · F + · G )

))= f

(g(φ(⟨ · F + · G ,d⟩))).

And henceξ ∈ supp(FG) ⇔⟨ξ,d⟩ = y + z,

where y ∈ supp( f ) ⊆ [−R,R] and z ∈ supp(g ) ⊆ R\] − R − ε,R + ε[. As a consequence ξ ∈supp(FG) if ⟨ξ,d⟩ ∈R\ [−ε,ε]. Hence 0 ∈ ξ ∈Rn : ⟨ξ,d⟩ ≤ ε 6⊆ supp(FG). Thus Theorem 2.2.13shows that R(FG) ∈S ′(Rn).

We will now show that F RG ∈S ′(Rn) is well defined. Theorem 2.2.13 shows that RG ∈S ′(Rn)is well defined. But then Theorem A.2.14 shows that F RG ∈S ′(Rn) is well defined.

Thus it remains only to show F RG = R(FG). To show this it is sufficient to show thatF RdG = Rd (FG) since by Theorem 2.2.15 Rl g , respectively Rl FG vanish for all l ∈Rn : ⟨l ,d⟩ = 0.Indeed

Rd FG −F RdG = Rd Ld ( f g )−Ld ( f )Rd Ld (g )

=Ld (H f g )−Ld ( f H g )

=Ld (H f g − f H g )

= 0,

where we used Theorem 2.2.18, Theorem 2.2.15 and Proposition 2.2.14.

Remark 2.2.20In [54] it was proven that a translation invariant operator on L2(Rn) satisfies a Bedrosian the-orem iff it is a linear combination of compositions of partial Hilbert transforms. That is theFourier multiplier of the operator is of the form

∑α

aαn∏

k=1

xαkk∣∣xαkk

∣∣ ,

where aα ∈C and α= (α1, . . . ,αn) ∈Nn0 is a multiindex. As a consequence a general Bedrosian

theorem for the Riesz transform can not exist.

EXAMPLE 2.2.3 (The Bedrosian identity and phase decomposition of signals):The analytical signal yields a unique decomposition of a function f = a cos(ϕ), where a is theamplitude and ϕ the phase of the analytical signal.

1. Let us now consider a function of the form

f (t ) = a(t )cos(ϕ(t )),


with a(t ),cos(ϕ(t )) ∈ R, ∀t ∈ R and∫R

a(t ) · (t )d t ,∫R

cos(ϕ(t )) · (t )d t ∈ S ′(R). Wewant to know under which conditions its analytical signal will be

fa(t ) = f (t )+ iH f (t ) = a(t )exp(iϕ(t )),

that is a is the amplitude and ϕ is the phase.

Theorem 2.2.18 tells us that this is the case, if ∃ −∞ < c < 0 < d < ∞ such thatsupp(a) ⊆ [c,d ] and supp(àcos(ϕ)) ⊂R\ [−d ,−c].

A possible choice is a(t ) = cos(r t ), ϕ(t ) = st , ∀t ∈R, where 0 < r < s <∞.

2. Theorem 2.2.19 answers a similar question for the Riesz transform respectively the mono-genical signal defined in chapter 4:

Considering a functionF (x) = A(x)cos(Φ(x))

such that A(x),cos(Φ(x)) ∈R, ∀x ∈Rn and A, cos(Φ) define a distribution in S ′(Rn).

Let

Fm(x) = (F (x),RF (x)) = A(

cos(φ),RF

|RF | sin(φ))∈Rn+1

define the monogenic signal, where

A(x) = |F (x),RF (x)|Rn+1

and

Φ(x) = arg( F (x)+ i |RF (x)||F (x),RF (x)|Rn−1

).

We want to know under which conditions A will be the amplitude andΦ the phase of themonogenic signal, i.e.

∃d(x) ∈Rn : Fm(x) = A(

cos(Φ(x))+d(x)sin(Φ(x))).

Theorem 2.2.19 states that this is the case if F := Ld ( f ) , where f = a cos(ϕ). It followsthat F is in distributional sense equal to F (x) = a(⟨x,d⟩)cos(ϕ⟨x,d⟩), whereA :=Ld (a) = a(⟨ · ,d⟩) is the amplitude andΦ=ϕ(⟨ · ,d⟩) is the phase.

Chapter 3

Clifford algebras andClifford modules

This chapter gives an introduction into Clifford algebras which is needed to understand theconcepts used in the following chapters. Section 3.1 states basic facts about Clifford algebraswhich are needed throughout the rest of this work. Section 3.2 introduces Clifford modules togive the setting in which we then introduce some already known facts about functional analysisin Clifford Hilbert modules. Finally in section 3.2.3 we will for the first time systematicallyconsider left-linear operators on Clifford-Hilbert modules. This results in new insights statedin Theorem 3.2.7 – Corollary 3.2.13. These insights will be essential for deriving uncertaintyrelations in chapter 6 and the development of Clifford frames in section 5.2.

The final part of the chapter treats some basics of Clifford analysis which will be useful in chap-ter 4.

3.1 Clifford algebra

In Theorem 2.1.7 we saw that the name analytical signal is due to a connection to complexanalysis. Thus, a complement to complex numbers and complex analysis for general dimen-sion would be desirable.

Generalisations of complex numbers are provided by Clifford algebras, which we will introducenow. The contents of this section is common knowledge about Clifford algebras found forexample in [21] and [19].

3.1.1 Definition of Clifford algebras

The complex numbersC areR2 equipped with an algebraic structure. We will now define analgebraic structure (i.e. a multiplication) onRn .

Definition 3.1.1 (Clifford algebra)LET n ∈N and e0,e1, . . . ,en be an OrthoNormal Basis (ONB) of Rn+1.

THEN a multiplication (a,b) → ab onRn ×Rn is defined by its action on the basis elements

(i) eαeα =−e0

41

42 CHAPTER 3. CLIFFORD ALGEBRAS

(ii) eαeβ =−eβeα, ∀α,β ∈ 1, . . . ,n : α 6=β.

The resulting algebra overRn respectivelyRn+1 is called Clifford algebra and denoted byRn .

Note that for every element α1, . . . ,αν of the power set P(1, . . . ,n), withν ∈ 1, . . . ,n elements, there exists by this definition an element eα1,...,αν := eα1 · · ·eαν . Linearcombinations of such elements are called ν-vectors. For the empty set ; the correspondingelement is denoted by e0 := e;. It follows that e0 is the unit element, i.e.,

e0eα = eαe0 ∀α ∈ 1, . . . ,n, and

e0e0 = e0.

This definition is clearly inspired by the construction of complex numbers. Indeed examplesfor Clifford algebras areR0 =R, R1 =C andR2 =H, (the Quaternions) which are the associa-tive division algebras. HoweverR3 6=O (the Octonions) which can be easily seen, as Cliffordalgebras are associative, whereas Octonions are not.

It should be noted that in a more general definition of Clifford algebra, item (i) in our definitionis replaced by a more general quadratic form. We will see later that our definition allows us todefine a Clifford analysis without encountering zero divisors.

3.1.2 Properties of Clifford algebras

We know that vectors are elements of a Clifford algebra. Let us now have a look at the generalelements of this algebra. To simplify notation we will use multivectors αν = α1, . . . ,αν, whereν ∈ 1, . . . ,n∪0 is the number of the elementsαk ∈ 1, . . . ,n. We will often skip the number ofelements, simply writing αν =α. The set of ordered elements of the product set P(1, . . . ,n)play a special role hence we will denote them as

On := αν ∈P(1, . . . ,n) : 1 ≤α1 <α2 < . . . <αν ≤ n; ν ∈ 1, . . . ,n∪ 0

. (3.1)

Let α,β ∈On then there exists s ∈ −1,1 and γ ∈On such that eγ = seαeβ. We will henceforthdenote sgn(αβ) := s.

The elements eαν may be visualised as oriented areas when ν= 2 and oriented volumes whenν= 3. By Definition 3.1.1(ii) the elements corresponding to the elements of the ordered subsetOn of the power set give a basis for the algebra regarded as a linear space. This basis consistsof 2n elements.

Hence a general element of a Clifford algebra can be written in this basis as

x = ∑αν∈Onν∈0,...,n

xανeαν =∑

α∈On

xαeα; xα ∈R.

We denote the part of an element of the Clifford algebra corresponding to the basis element eβ(β ∈On) by

⟨x⟩β = ⟨ ∑α∈On

xαeα⟩β = xβ ∈R.

The element corresponding to the empty set e0 plays a special role, since e0eα = eα = eαe0 andhence e0e0 = e0, it holds true that ae0x = ∑

α∈Onaxαeα. Thus, e0 is the identity of the algebra,

and multiplication with elements of spane0 is equal to scalar multiplication. Hence, spane0may be identified withR.

3.1. CLIFFORD ALGEBRA 43

3.1.3 Zero divisors

The Clifford algebras R0 =R, R1 =C and R2 =H are division algebras and thus posses nozero divisors.

EXAMPLE 3.1.1 (Zero divisors):For n ≥ 3, the element 1

2 (1+ e123) is an idempotent, and the corresponding zero divisor is (1+e123)(1−e123) = 0.

Proof. To see this let us first compute e2123:

e123e123 =−e123e1e3e2 = e12e3e3e12 = e12e21 =−e21 = 1.

Thus we see that

1

4(1+e123)(1+e123) = 1

4(1+2e123 +e2

123) = 1

2(1+e123),

respectively that(1+e123)(1−e123) = 1−e123 +e123 −e2

123 = 0.

3.1.4 Paravectors and the paravector group

Definition 3.1.2 (Paravectors and paravector group)LET n ∈N, andRn the corresponding Clifford algebra.

THEN elements of the form

a0 +~a :=n∑α=0

aαeα

are called paravectors. a0 is called the real part or scalar part, and ~a the vector part of theparavector. The paravectors form the n +1-dimensional linear subspaceRn+1 of the Cliffordal-gebraRn .

The group a ∈Rn : a = ∏l al , al ∈Rn+1 spanned by the paravectors under multiplication is

called the paravector group.

Theorem 3.1.3 (The paravector group and zero deivisors)LET n ∈N.

THEN the paravector group ofRn is free of zero divisors.

Proof. For a proof, see [19] Corollary (5.20).

3.1.5 Scalar product, Euclidean measure and conjugation

Definition 3.1.4 (A scalar product onRn)The Euclidean scalar product of the embedding ofRn intoR2n

defines a scalar product (·, ·) onRn :

(a,b) =( ∑α∈On

aαeα,∑

α∈On

bαeα

):= ∑

α∈On

aαbα


This scalar product induces a norm | · | = (·, ·)1/2 which is called modulus.

Corollary 3.1.5(Rn , (·, ·)) is a real Hilbert space.

Theorem 3.1.6 (Conjugation of paravectors)A conjugation is defined on the paravectors by

:Rn+1 →Rn+1; x = x0 +n∑α=1

eαxα 7→ x0 −n∑α=1

eαxα =: x,

This conjugation fulfills

(x, y) =n∑α=0

xαyα = ⟨x y

⟩0 =

1

2(x y + y x), ∀x, y ∈Rn+1

and(x, x) = xx = |x|2, ∀x ∈Rn+1.

As a consequence (x, y) = (x, y), ∀x, y ∈Rn+1.

Proof. Let x, y ∈Rn+1.

x y = x0 y0 −x0∑α

eαyα+ y0∑α

eαxα−∑α,β

eαβxαyβ

= x0 y0 −x0∑α

eαyα+ y0∑α

eαxα+∑α

xαyα−∑α<β

eαβ(xβyα−xαyβ)

and

x y = x0 y0 +x0∑α

eαyα− y0∑α

eαxα−∑α,β

eαβxαyβ

= x0 y0 +x0∑α

eαyα− y0∑α

eαxα+∑α

xαyα−∑α<β

eαβ(xβyα−xαyβ).

Hence

x y + y x = 2x0 y0 −∑α<β

eαβ((xβyα−xαyβ)− (yβxα− yαxβ)

)+2∑α

xαyα

= 2(x, y).

Corollary 3.1.7 (Inverses of paravectors)LET x ∈Rn+1 be a paravector.

THEN x is invertible with inverse x|x|2 .

Proof. Note that x x|x|2 = (x,x)

|x|2 = 1.

Definition 3.1.8 (Hypercomplex Conjugation)The hypercomplex conjugation on arbitrary elements of the Clifford algebra is defined by

eα =−eα ∀α ∈ 1, . . . ,n

ab = ba ∀a,b ∈Rn

3.1. CLIFFORD ALGEBRA 45

For the ν-vectors eαν , αν ∈On , ν ∈ 1, . . . ,n of the canonical basis the conjugate is the inverse:

eα1,...,ανeα1,...,αν = eα1,...,αν−1 eαν (−eαν )eα1,...,αν−1 = . . . = 1 ∀ν ∈ 1, . . . ,n.

Corollary 3.1.9LET eαν , αν ∈P(1, . . . ,n), ν≤ n.

THEN the conjugation of eαν is equal to a change in sign:

eαν = (−1)ν(ν+1)

2 eαν .

As a consequence eαν = eαν .

Proof. eαν = eαν . . .eα1 = (−1)νeαν . . .eα1 = (−1)ν(ν−1)

2 (−1)νeαν .

Using the conjugation we define aRn-valued product.

Definition 3.1.10 (TheRn-valued product⟨⟨ · , · ⟩⟩

)TheRn-valued product

⟨⟨ · , · ⟩⟩is defined by⟨⟨·, ·⟩⟩ :Rn ×Rn →Rn ,

⟨⟨x, y

⟩⟩:= x y .

Corollary 3.1.11LET a ∈Rn+1 be an element of the paravector group.

THEN aa = ⟨⟨a, a

⟩⟩= (a, a) = |a|2. Furthermore a is invertible with the inverse a(a,a) .

Proof. We know that this is true for paravectors. Thus for a = ∏lk=1 ak 6= 0, where ak 6= 0 are

paravectors, and ⟨⟨a, a

⟩⟩= aa = a1 . . . al al︸︷︷︸∈R+

. . . a1 =l∏

k=1(ak , ak ) ∈R+.

It follows that⟨⟨

a, a⟩⟩= (a, a).

3.1.6 Complex Clifford algebras

We will consider the complexificationCn∼=C⊗Rn of the Clifford algebraRn given by

z = a + i b = ∑α∈On

aαeα+ i∑

α∈On

bαeα = ∑α∈On

zαeα, zα ∈C, aα,bα ∈R.

The real part of an element of a complex Clifford algebra will be defined as ℜz =ℜ(⟨z⟩0).

Note that 12 (1± i eα) is an idempotent for all α ∈ 1, . . . ,n, and hence a zero divisor. (Note that

(i eα)2 = 1).

Conjugation is defined as the combination of complex and hyper-complex conjugations

eα =−eα, ∀α ∈ 1, . . . ,n, i =−i and ab = ba, ∀a,b ∈Cn .

A scalar product (·, ·) is defined by the Euclidean scalar product via the embedding ofCn into

R2n+1.


Corollary 3.1.12LET z = a + i b, x = l + i r , where a,b, l ,r ∈Rn .

THEN

(i) (x, z) =ℜ(xz).

(ii) (x y, z) = (x, z y) ∀y ∈Cn .

Proof. ad(i)

ℜ(xz) =⟨ ∑α,β∈On

eαeβℜ(xαzβ)⟩

0=

⟨ ∑α,β∈On

eαeβ(lαaβ+ rαbβ)⟩

0= ∑α∈On

aαlα+bαrα

= (x, z).

ad(ii) (x y, z) =ℜ(x y z) =ℜ(x y z) =ℜ(xz y) = (x, z y).

3.2 Hypercomplex functional analysis

3.2.1 Clifford modules

We will now define Clifford modules especially over Hilbert spaces. The ultimate goal here is togive a setting in which frames on Clifford modules (especially on the module over L2(Rn ,Rn))may be defined.

The first two sections are based on the according results in [32] and [7]. Theorem 3.2.4 is acorrected and expanded version of [32] Proposition 1.9.

The final section 3.2.3 consists of novel facts on left-linear operators on Clifford-Hilbert mod-ules.

Definition 3.2.1 (Clifford module)LET V be aK-vector space, whereK=R orK=C.

THEN the corresponding two-sided K Clifford module is defined as

Vn :=V ⊗Rn :=

x = ∑α∈On

xα⊗eα; xα ∈V

andKn acts on Vn by

ax := ∑α,β∈On

aβxα⊗eβeα, xa := ∑α,β∈On

aβxα⊗eαeβ; ∀x ∈Vn , a ∈Kn .

We will define the α-part of an element x ∈Vn to be

⟨x⟩α =⟨ ∑β∈On

xβ⊗eβ⟩α

:= xα.

The real part of an x ∈Vn will be defined as

ℜx =ℜx0 ⊗e0.

3.2. HYPERCOMPLEX FUNCTIONAL ANALYSIS 47

If (V ,‖ ·‖V ) is a normed vector space then Vn can be equipped with the Euclidean norm∥∥∥∥∥ ∑α∈On

xα⊗eα

∥∥∥∥∥Vn

:=( ∑α∈On

‖xα‖2V

)1/2

.

If W ⊆Vn is a left-(right-) submodule of Vn then a left (right)Cn-linear mapping

L : W 7→Cn

is called a Clifford functional of W. The collection of all bounded Clifford functionals of W iscalled the dual Clifford module and is denoted by W ∗.

Note that l : Vn 7→Cn is called left-linear iff l (ax) = al (x) (respectively right linear l (xa) = l (x)afor right-modules).

In the following we will often write x =∑α∈On

xαeα :=∑α∈On

xα⊗eα for an element x ∈Vn .

Proposition 3.2.2 (Clifford functionals and functionals)LET V be a complex normed vector space.

(i) LET l : Vn 7→C be a linear functional.

THEN l (x) := ∑α∈On

l (eαx)eα is a (left-linear) Clifford functional. Furthermore l = ⟨l⟩0.More generally, for every α ∈ On there exists a uniquely defined (left-)linear functionall[α](x) = l (eαx) such that l (x) = ⟨l[α](x)⟩α.

(ii) LET h : Vn 7→Cn be a Clifford functional.

THEN l = ⟨h⟩0 is aC-linear functional and l = h.

(iii) LET h be a (left-)Clifford functional.

THEN h = 0 ⇔∃α ∈On : ⟨h⟩α = 0.

(iv) h is a bounded Clifford functional iff ⟨h⟩α is a bounded functional for some α ∈On .

Proof. For the proof we will follow [7].

ad(i) By construction it is clear that l isC-linear and that l = ⟨l⟩0. We need to show that

l (ax) = al (x), ∀a ∈Cn .

Let a =∑β∈On

aβeβ =∑nν=0

∑βν∈On|βν|=ν

aβνeβν . Then

l (ax) = ∑α∈On

l (eαax)eα = ∑α,β∈On

aβl (eαeβx)eα

∗=n∑ν=0

∑α,βν∈On

(−1)ν(ν+1)

2 aβν l (eβνeαx)eβνeβνeα

γ=βνα=n∑ν=0

∑γ,βν∈On

aβν l (eγx)(−1)ν(ν+1)

2 eβνeγ

= ∑γ,β∈On

aβl (eγx)eβeγ

= al (x), ∀x ∈Vn .


To see * note that βν ∈β ∈P(1, . . . ,n) : 1 ≤ β1 < β2 < . . . < βν ≤ n

= β ∈On : |β| = ν.By Definition 3.1.8 and Corollary 3.1.9

eαeβν = eα(−1)ν(ν+1)

2 eβν = (−1)ν(ν+1)

2 eβνeα.

Furthermore if we set l[α](x) := l (eαx) =∑β∈On

eβl (eαeβx), then l[α] isCn-linear by con-

struction. Hence ⟨l (eαx)⟩α = ⟨eα l (x)⟩α = ⟨l (x)⟩0 = l (x).

We will now proof the uniqueness of l[α]. For any α ∈ On and a ∈Cn let the C-linearfunctional ta,α be defined by

ta,α :Cn 7→C; b 7→ ta,α(b) := l (eαba).

Then by the Riesz representation theorem (see for example [56] V.3.6) onCn , interpretedas the complex Hilbert space

(C2n

, ( · , · )), there exists a unique µa,α ∈Cn such that

ta,α(b) = (b,µa,α).

Putting b = eα we obtain

⟨µa,α⟩α = (eα,µa,α) = ta,α(eα) = l (eαeαa) = l (a).

If we defineλ[α](a) :=µa,α,

then⟨λ[α](a)⟩α = l (a).

Furthermore λ[α] isCn-linear as seen by the definition using Corollary 3.1.12: Let c,d ∈Cn .

(b,λ[α](c +d a)) = (b,µc+d a,α) = tc+d a,α(b) = l (eαb(c +d a))

= l (eαbc)+ l (eα(bd)a) = (b,λ[α](c))+ (bd ,λ[α](a))

= (b,λ[α](c)+dλ[α](a)).

Now l (eαa) =λ[α](a) since(b, l[α](a)

)= ⟨b

∑β∈On

eβ⟨l[α](a)⟩β⟩

0= ∑β∈On

bβl (eαeβa)

= l (eαba) = ta,α(b) = (b,µa,α) = (b,µa,α)

= (b,λ[α](a)

) ∀b ∈Cn ,

whence l[α] is unique.

ad(ii) We will first show that l isC-linear. Let z ∈C∼=Ce0 then

l (zx) = ⟨h(zx)⟩0 = ⟨zh(x)⟩0 = zl (x), ∀x ∈Vn .

Now

l (x) = ∑α∈On

l (eαx)eα = ∑α∈On

⟨h(eαx)⟩0eα = ∑α∈On

⟨eαh(x)⟩0eα

= ∑α∈On

⟨ ∑β∈On

eαeβ⟨h(x)⟩β⟩

0

eα = ∑α∈On

⟨h(x)⟩αeα

= h(x), ∀x ∈Vn .


ad(iii) Let h be a Clifford functional. If h = 0 then ⟨h⟩α = 0, ∀α ∈On .

Conversely suppose hα = 0 for some α ∈ On . Then h is the unique Clifford functionalsuch that

(⟨h⟩α)[α] = h. Hence by the proof of Proposition 3.2.2(i)

h(a) = ∑β∈On

eβ⟨h(eβa)⟩0 =∑β∈On

eβ⟨h(eαeβa)⟩α = 0.

ad(iv) Let h ∈V ∗n . Then, ⟨h⟩α is bounded for all α ∈On . Conversely let α ∈On and let

|⟨h(a)⟩α| ≤C |a|, ∀a ∈Vn .

Then

|h(a)|2 = ∑β∈On

∣∣⟨h(a)⟩β

∣∣2 = ∑β∈On

∣∣⟨h(eβa)⟩0

∣∣2 = ∑β∈On

∣∣⟨h(eαeβa)⟩α

∣∣2

≤ ∑β∈On

C 2|eαeβa|2 ≤ ∑β∈On

C 2|eα|2|eβa|2

≤ 2nC 2|a|2.

Theorem 3.2.3 (The Hahn-Banach Theorem )LET V be a complex normed vector space and let Xl be a left-submodule of Vn . Furthermorelet h be a left-linear Clifford functional on Xl .

THEN there exists a left-linear Clifford functional h∗ on Vn such that

h∗∣∣Xl

= h and ‖h‖ ≤ ‖h∗‖ ≤ 2n/2‖h‖.

Proof. The theorem was stated in [32] without a proof. Hence we will give the proof.

By the complex case of the Hahn-Banach theorem (see for example [56] III.1.5) we can extend⟨h⟩0 to a complex linear functional on Vn denoted by ⟨h∗⟩0 of the same norm‖⟨h⟩0‖ = ‖⟨h∗⟩0‖.

By Proposition 3.2.2(ii) we know that ⟨h∗⟩0∣∣

Xl= h. If we denote h∗ := ⟨h∗⟩0, then ‖⟨h∗⟩0‖ =

‖⟨h⟩0‖ is bounded, whence ‖h‖ ≤ ‖h∗‖ and by Proposition 3.2.2(iv) ‖h∗‖ ≤ ‖⟨h⟩0‖ ≤ 2n/2‖h‖.

3.2.2 Clifford-Hilbert modules

Let(H ,⟨ · , · ⟩H

)be a complex Hilbert space. We will now consider Hn := H ⊗Cn the space,

that has elements of the form x =∑α∈On

xαeα, where xα ∈ H , ∀α ∈On . Hn is a complex Hilbertspace when endowed with the scalar product

( · , · ) : Hn ×Hn →C, (x, y) := ∑α∈On

⟨xα, yα⟩H , ∀x, y ∈ Hn .

This scalar product induces the Euclidean metric

‖x‖2Hn

:= (x, x) = ∑α∈On

‖xα‖2H .


The Euclidean scalar product just introduced is not Clifford algebra-valued. Inspired by thescalar product onCwe will now introduce a complexified Clifford algebra-valued form on Hn .⟨⟨ · , · ⟩⟩

:Hn ×Hn →Cn ,⟨⟨

x, y⟩⟩

:= ∑α,β∈On

⟨xα, yβ⟩H eαeβ

If is a conjugation on H , then

: Hn 7→ Hn , x 7→ ∑α∈On

xα eα

defines a conjugation on Hn .

Theorem 3.2.4 (Properties of the Clifford algebra-valued inner product)LET x, y ∈ Hn and a ∈Cn .

THEN the following hold.

(i)⟨⟨

ax, y⟩⟩= a

⟨⟨x, y

⟩⟩,

⟨⟨x, ay

⟩⟩= ⟨⟨x, y

⟩⟩a,

⟨⟨xa, y

⟩⟩= ⟨⟨x, y a

⟩⟩and

⟨⟨x, y

⟩⟩= ⟨⟨y, x

⟩⟩.

(ii)⟨⟨

x, y⟩⟩=∑

α∈On(eαx, y)eα, in particular,

⟨⟨⟨x, y

⟩⟩⟩0 = (x, y).

(iii)∣∣⟨⟨x, y

⟩⟩∣∣≤ 2n/2‖x‖‖y‖.

(iv) ℜ⟨⟨x, x

⟩⟩= ‖x‖2. In particular, ‖x‖2 ≤ ∣∣⟨⟨x, x⟩⟩∣∣≤ 2n‖x‖2.

(v) ‖ax‖ ≤ 2n/2|a|‖x‖. However if a ∈Rn+1 is a paravector, or an element of the paravectorgroup, then ‖ax‖ = |a|‖x‖.

(vi) x = x, ax = x a, xa = a x.

(vii) ‖x‖ ≤ sup‖y‖≤1

∣∣⟨⟨x, y⟩⟩∣∣≤ 2n/2‖x‖.

Proof. (i)–(vi) can be found in [32], while (vii) seems to be new.

ad(i) ⟨⟨ax, y

⟩⟩= ∑α,β,γ∈On

⟨aαxβ, yγ⟩H eαeβeγ = ∑α∈On

aαeα∑

β,γ∈On

⟨xβ, yγ⟩H eβeγ = a⟨⟨

x, y⟩⟩

.

⟨⟨x, ay

⟩⟩= ∑α,β,γ∈On

⟨xα, aβyγ⟩H eαeβeγ = ∑α,γ∈On

⟨xα, yγ⟩H eαeγ∑β∈On

aβeβ = ⟨⟨x, y

⟩⟩a.

⟨⟨xa, y

⟩⟩= ∑α,β,γ∈On

⟨xαaβ, yγ⟩H eαeβeγ = ∑α,β,γ∈On

aβ⟨xα, yγ⟩H eαeβ eγ

= ∑α,β,γ∈On

⟨xα, yγaβ⟩H eαeγeβ = ⟨⟨x, y a

⟩⟩.

⟨⟨x, y

⟩⟩= ∑α,β∈On

⟨xα, yβ⟩H eαeβ = ∑α,β∈On

⟨yβ, xα⟩H eβeα = ⟨⟨y, x

⟩⟩.

ad(ii) We easily see that by definition (., .) = ⟨⟨⟨., .⟩⟩⟩0. Now (i) shows that

⟨⟨., y

⟩⟩suffices the

requirements of Proposition 3.2.2(ii) which proves (ii).


ad(iii)

|⟨⟨x, y⟩⟩|2 (i i )=

∣∣∣ ∑α∈On

(eαx, y

)eα

∣∣∣2 = ∑α∈On

∣∣(eαx, y)∣∣2 ∗≤ ∑

α∈On

‖eαx‖2‖y‖2

= 2n‖x‖2‖y‖2,

where in ∗ the Cauchy-Schwartz-inequality in H has been used.

ad(iv) ‖x‖2 = (x, x) = ⟨⟨⟨x, x

⟩⟩⟩0 = ℜ⟨⟨

x, x⟩⟩

. Hence ‖x‖2 ≤ |⟨⟨x, x⟩⟩|. Now |⟨⟨x, x

⟩⟩| ≤ 2n‖x‖2

follows from (iii).

ad(v) Let a ∈Rn+1, then

‖xa‖2 (i v)= ℜ⟨⟨xa, xa

⟩⟩ (i )= ℜ⟨⟨xaa, x

⟩⟩ ∗= |a|2ℜ⟨⟨x, x

⟩⟩= |a|2‖x‖2.

In ∗, we used the fact that for paravectors aa = |a|2 ∈R, hence the assumption ⟨a, a⟩ =|a|2 is sufficient for the equality to hold.

By definition it is clear that |a| = |a|, ∀a ∈Cn , and ‖x‖ = ‖x‖, ∀x ∈ Hn . Hence ‖ax‖ =‖x a‖ = ‖x a‖ = |a|‖x‖ = |a|‖x‖.

Let now a ∈Cn . Then

‖ax‖2 =∥∥∥ ∑α,β∈On

aαxβeαeβ∥∥∥2 ≤

( ∑α∈On

|aα|∥∥∥ ∑β∈On

xβeαeβ∥∥∥)2

∗≤ ∑α∈On

|aα|2∑γ∈On

∥∥∥ ∑β∈On

xβeγeβ∥∥∥2

= 2n |a|2‖x‖2.

In * we used the Cauchy-Schwartz inequality.

ad(vi) x = x is clear since this holds for the conjugation on H andCn . Now

ax = ∑α,β∈On

aαxβeαeβ = ∑α,β∈On

xβ aα eβ eα = x a.

xa = a x follows in the same way.

ad(vii) Since(Hn , (·, ·)) is a complex Hilbert space

‖x‖ = sup‖y‖≤1

|(x, y)| ≤ sup‖y‖≤1

|⟨⟨x, y⟩⟩|.

By (iii) sup‖y‖≤1 |⟨⟨

x, y⟩⟩| ≤ 2n/2‖x‖.

EXAMPLE 3.2.1 (Examples for Theorem 3.2.4(iii)):We will now give examples for Theorem 3.2.4(iii) which demonstrate in which cases there isequality and in which cases the inequality holds without the factor 2n/2.

(i) LET x, y ∈ Hn : (eαx, y) = ‖x‖‖y‖, ∀α ∈On .

THEN by Theorem 3.2.4(ii)∣∣⟨⟨x, y⟩⟩∣∣2 = ∑

α∈On

|eα(eαx, y)|2 = ∑α∈On

‖x‖2‖y‖2 = 2n‖x‖2‖y‖2.


(ii) LET x, y ∈ Hn : ⟨xα, yβ⟩H = 0, ∀α 6=β ∈On .

THEN∣∣⟨⟨x, y⟩⟩∣∣2 = ∑

α,β∈On

|⟨xα, yβ⟩|2 =∑

α∈On

|⟨xα, yα⟩|2 ≤∑

α∈On

‖xα‖2‖yα‖2 ≤ ∑α∈On

‖xα‖2∑β

‖yβ‖2

= ‖x‖2‖y‖2.

Theorem 3.2.5 (Riesz representation theorem)LET V be a left-submodule of Hn and L ∈V ∗ a Clifford functional.

THEN there exists a unique element a ∈ V such that L(x) = ⟨⟨x, a

⟩⟩. Moreover ‖a‖ ≈ ‖L‖. The

operatorΦ : V 7→V ∗, a 7→ ⟨⟨·, a⟩⟩

is bijective and conjugate linear.

Proof. This theorem can be found in [32]. By the Riesz representation theorem for complexHilbert spaces (see for example [56] V.3.6), ∃a ∈V : ⟨L(.)⟩0 = (., a) = ⟨⟨⟨

., a⟩⟩⟩

0. Let x ∈ Hn . Then

L(x) = ∑α∈On

⟨eαL(x)⟩0eα = ∑α∈On

⟨L(eαx)⟩0eα

= ∑α∈On

⟨⟨⟨eαx, a

⟩⟩⟩0eα = ∑

α∈On

⟨eα

⟨⟨x, a

⟩⟩⟩0eα

= ⟨⟨x, a

⟩⟩.

The representation is unique since⟨⟨

x, a⟩⟩= 0, ∀x ∈V ⇒ ⟨⟨⟨

a, a⟩⟩⟩

0 = ‖a‖ = 0, and hence a = 0.

Now the equivalence of the norms is seen by ‖a‖2 = ⟨⟨⟨a, a

⟩⟩⟩0 ≤

∣∣⟨⟨a, a⟩⟩∣∣ whence

‖a‖ ≤∣∣∣⟨⟨ a

‖a‖ , a⟩⟩∣∣∣≤ sup

‖y‖=1|⟨⟨y, a

⟩⟩| = sup‖y‖=1

|L(y)| = ‖L‖,

and sup‖y‖=1 |L(y)| = sup‖y‖=1 |⟨⟨

y, a⟩⟩| 3.2.4(i i )≤ 2n/2‖a‖.

3.2.3 Operators on Clifford-Hilbert modules

In the following section we will consider left-linear operators on Clifford modules. It seemsthat this is the first time that these operators are considered systematically, whence all resultshave to be considered novel. However many results correspond quite closely to the results onlinear operators on Hilbert spaces. The results which differ from the Hilbert space case areTheorem 3.2.7(ix), Corollary 3.2.8 and the results on self-adjoint operators Proposition 3.2.10and Theorem 3.2.12.

Let L(Hn ,Gn) be the module of Clifford left-linear bounded operators from the Clifford-Hilbertmodule Hn to the Clifford-Hilbert module Gn .

For an operator T ∈ L(Hn ,Gn) an adjoint in the sense of Clifford modules of Banach spaces isdefined by

T ′ : G∗n 7→ H∗

n , y∗ 7→ T ′(y∗) : x 7→ T ′(y∗)(x) := y∗(T x).

Definition 3.2.6 (The adjoint operator)LET T ∈ L(Hn ,Gn).

THEN its adjoint is defined by

T ∗ : Gn 7→ Hn ;⟨⟨

T x, y⟩⟩

Gn= ⟨⟨

x,T ∗y⟩⟩

Hn.

By Theorem 3.2.5 this is equal to T ∗ =Φ−1Hn

T ′ΦGn .


Theorem 3.2.7 (Operators and their adjoints)LET Hn ,Gn ,Fn be Clifford-Hilbert modules. Furthermore let S,T ∈ L(Hn ,Gn),R ∈ L(Gn ,Fn), and a ∈Cn .

THEN

(i) (S +T )∗ = S∗+T ∗.

(ii) (Sa)∗ = S∗a.

(iii) (RS)∗ = S∗R∗.

(iv) S∗ ∈ L(G∗n , H∗

n ).

(v) S∗∗ = S.

(vi) ‖S‖ = ‖S∗‖.

(vii) ‖SS∗‖ = ‖S∗S‖ = ‖S‖2.

(viii) Ker(S) = Ran(S∗)⊥; Ker(S∗) = Ran(S)⊥.

(ix) there exists a set of linear operators

Sββ∈On, Sβ : H →G

such thatS f = ∑

α,β∈On

eαeβSβ fα.

(Keep in mind that in order to be left-linear the operator S =∑α∈On

eαSα has to be appliedfrom the right.)

The operator S is bounded iff for every α ∈On the operator Sα is bounded.

Proof.

ad(i)⟨⟨

x, (S +T )∗y⟩⟩= ⟨⟨

(S +T )x, y⟩⟩= ⟨⟨

Sx, y⟩⟩+⟨⟨

T x, y⟩⟩= ⟨⟨

x,S∗y⟩⟩+⟨⟨

x,T ∗y⟩⟩

ad(ii)⟨⟨

(Sa)x, y⟩⟩= ⟨⟨

Sxa, y⟩⟩= ⟨⟨

xa,S∗y⟩⟩ 3.2.4(v)= ⟨⟨

x,S∗y a⟩⟩= ⟨⟨

x, (S∗a)y⟩⟩

ad(iii)⟨⟨

RSx, y⟩⟩

Hn= ⟨⟨

Sx,R∗y⟩⟩

Gn= ⟨⟨

x,S∗R∗y⟩⟩

Fn

ad(iv) We will show S∗ ∈ L(G∗n , H∗

n ).

S∗ is Clifford linear, since⟨⟨

x,S∗(y +az)⟩⟩= ⟨⟨

Sx, y⟩⟩+⟨⟨

Sx, z⟩⟩

a = ⟨⟨x,S∗y +aS∗z

⟩⟩.

Boundedness follows since

‖S∗‖2 = sup‖x‖=1

(S∗x,S∗x) = ⟨⟨⟨x,SS∗x⟩⟩⟩0 = (x,SS∗x)

∗≤ sup‖x‖=1

‖x‖‖SS∗x‖ ≤ sup‖x‖=1

‖S∗‖‖Sx‖‖x‖

= ‖S∗‖‖S‖,

where ∗ follows from the Cauchy-Schwartz inequality on the Hilbert space(Hn , (·, ·)).


ad(v) By (iv) S∗∗ ∈ L(H∗∗n ,G∗∗

n ). Hence (v) is obvious, since⟨⟨S∗∗x, y

⟩⟩G∗∗

n= ⟨⟨

x,S∗y⟩⟩

Hn= ⟨⟨

Sx, y⟩⟩

Gn

and G∗∗n =Gn .

ad(vi) Using (iv) and (v) ‖S‖ ≥ ‖S∗‖ ≥ ‖S∗∗‖ = ‖S‖.

ad(vii) Is obvious since it is true in the complex Hilbert space(Hn , (·, ·)).

ad(viii) Let x ∈ Ker(S). Then

Sx = 0 ⇔⟨⟨Sx, y

⟩⟩= 0 ∀y ∈Gn

⇔⟨⟨x,S∗y

⟩⟩= 0 ∀y ∈Gn

⇔x ∈ (Ran(S∗)

)⊥.

Furthermore Ker(S∗) = (Ran(S∗∗)

)⊥ = (Ran(S)

)⊥.

ad(ix) Let S =∑α∈On

eαSα : Hn →Gn , f =∑β∈On

eβ fβ 7→∑α,β∈On

eβeαSα fβ. Furthermore leta ∈Cn , f , g ∈ Hn . Then

S(a f + g ) = ∑α,β

eβeαSα(⟨a f ⟩β+⟨g ⟩β) = ∑α,β

eβeα(Sα

( ∑σ,τ∈On

eσeτ=±eβ

sgn(στ)aσ fτ)+Sα

(⟨g ⟩β))

= ∑α,σ,τ∈On

eσeτeαaσSα( fτ)+ ∑α,β∈On

eβeαSα(gβ)

= aS( f )+S(g ),

where eβ = sgn(στ)eσeτ. Hence S is a left-linear operator on the Hilbert Clifford mod-ule.

Let us now assume that S is a left-linear operator on the Hilbert Clifford module. Foran element a =∑

α∈Oneαaα of the Clifford module denote

aβ :=− ∑α∈Onα6=β

eαaα+eβaβ.

Then a +aα = 2eα⟨a⟩α and consequently a = 12

∑α∈On

a +aα. Furthermore, since S isleft-linear

S( f ) = S(1

2

∑α∈On

f + fα

)= 1

2

∑α∈On

S(

f + fα

)= 1

4

∑α,β∈On

(S +Sα

)(f + f

β

)= 1

4

∑α,β∈On

eα⟨

S(eβ⟨ f ⟩β

)⟩α= ∑α,β∈On

eα⟨

eβS(e0⟨ f ⟩β

)⟩α

= ∑α,β∈On

sgn(βγ)eα⟨S(e0⟨ f ⟩β

)⟩γ

,

where γ ∈ On : sgn(βγ)eβeγ = eα and sgn(βγ) is the sign such that sgn(βγ)eβeγ = eαfor some α ∈On .

Finally for f ∈ H and α ∈On denote Sα( f ) := ⟨S(e0 f )⟩α.


Suppose now ∀α ∈On∃Cα ∈R+ : ‖Sαg‖2 ≤Cα‖g‖2, ∀g ∈ H . Then for all f ∈ Hn

‖S f ‖2 = ∑α∈On

∥∥⟨S( f )⟩α∥∥2 = ∑

α∈On

∥∥ ∑τσ=α

Sσ( fτ)∥∥2

≤ ∑α∈On

∑τσ=α

∥∥Sσ( fτ)∥∥2 ≤ ∑

α∈On

∑τσ=α

Cσ

∥∥ fτ∥∥2 ≤C

∑τ‖ fτ‖2

=C‖ f ‖2,

for some C ∈R+.

On the other hand assume ∃α ∈ On , gl l∈N ⊂ H , ‖gl‖ = 1 such that ‖Sα(gl )‖l∈N isunbouded.

Let fl := gl e0. Then‖S fl‖ =

∑β∈On

‖Sβ(e0gl )‖2 ≥ ‖Sα(g )‖2,

whence the series ‖S fl‖l∈N is unbounded. Since ‖S‖ would be an upper bound forthis series, S is unbounded.

Corollary 3.2.8 (The adjoint operator)LET T ∈ L(Hn ,Gn).

THEN its adjoint T ∗ with respect to the scalar product (·, ·) is equal to its adjoint with respect tothe form

⟨⟨·, ·⟩⟩.

(T x, y) = (x,T ∗y), ∀x, y ∈ Hn ⇔ ⟨⟨T x, y

⟩⟩= ⟨⟨x,T ∗y

⟩⟩, ∀x, y ∈ Hn .

Proof. Let T ∈ L(Hn ,Gn) and fix y ∈Gn . Then h(x) := ⟨⟨T x, y

⟩⟩is a Clifford functional. As a con-

sequence l (x) = ⟨⟨h(x)

⟩⟩0 is aC-linear functional. Now l (x) = (T x, y) = (x,T ∗y). Furthermore

g (x) := ⟨⟨x,T ∗y

⟩⟩is a Clifford functional and ⟨g (x)⟩0 = (x,T ∗y) = l (x). By Proposition 3.2.2(ii)

we know that g (x) = l (x) = h(x).

Self-adjoint operators on Hilbert modules

Definition 3.2.9 (Unitary and self-adjoint operators)

• LET T ∈ L(Hn ,Gn).

THEN T is called unitary iff T T ∗ = IdGn , T ∗T = IdHn .

• LET T ∈ L(Hn) := L(Hn , Hn).

THEN T is called self-adjoint iff T = T ∗.

Proposition 3.2.10 (Characterization of self-adjoint operators)LET T ∈ L(Hn).

THEN the following are equal.

(i) T is self-adjoint.


(ii)⟨⟨

T x, x⟩⟩= ⟨⟨

T x, x⟩⟩

, ∀x ∈ Hn .

(iii) For any α : eα = eα the operator Tα ∈ L(H) is self-adjoint. For any α : eα =−eα the opera-tor i Tα ∈ L(H) is self-adjoint.

Proof. Let x, y ∈ Hn .

(i)⇒ (ii)⟨⟨

T x, x⟩⟩= ⟨⟨

x,T ∗x⟩⟩= ⟨⟨

x,T x⟩⟩= ⟨⟨

T x, x⟩⟩

.

(ii)⇒ (i) Let a ∈Cn . ⟨⟨T (x +ay), x +ay

⟩⟩= ⟨⟨T x, x

⟩⟩+⟨⟨T x, y

⟩⟩a +a

⟨⟨T y, x

⟩⟩+a⟨⟨

T y, y⟩⟩

a

(i i )=⇒ ⟨⟨T (x +ay), x +ay

⟩⟩= ⟨⟨T x, x

⟩⟩+a⟨⟨

y,T x⟩⟩+⟨⟨

x,T y⟩⟩

a +a⟨⟨

T y, y⟩⟩

a.

Applying (ii) yields ⟨⟨T x, y

⟩⟩a +a

⟨⟨T y, x

⟩⟩= a⟨⟨

y,T x⟩⟩+⟨⟨

x,T y⟩⟩

a.

Choosing a = 1 we see that⟨⟨T x, y

⟩⟩+⟨⟨T y, x

⟩⟩= ⟨⟨y,T x

⟩⟩+⟨⟨T x, y

⟩⟩. (3.2)

Now we use the fact that Hn is a sub-module of every Clifford-Hilbert module Hv , suchthat n ≤ v ∈N. We choose ν ∈N such that ν and v(v+1)

2 are odd. Then eν := e1...v is anelement of the commutator of the Clifford algebraCv , and eν =−eν. Choosing a = eν wehave

eν⟨⟨

y,T x⟩⟩−eν

⟨⟨x,T y

⟩⟩= eν⟨⟨

T y, x⟩⟩−eν

⟨⟨T x, y

⟩⟩. (3.3)

Now by multiplying (3.3) by eν from the left, and adding (3.2), (i) follows.

(iii)⇒ (ii) ⟨⟨T x, x

⟩⟩= ∑α,β,γ∈On

eαeβeγ⟨Tβxα, xγ⟩

(i i i )= ∑α,β,γ∈On

eαeβ eγ⟨xα,Tβxγ⟩ =⟨⟨

x,T x⟩⟩

= ⟨⟨T x, x

⟩⟩(ii)⇒ (iii) By Theorem 3.2.7(viii) ∀α ∈On∃Tα ∈ L(H) : T =∑

α∈OneαTα.

⟨⟨T x, x

⟩⟩= ∑α,β,γ∈On

eαeβeγ⟨Tβxα, xγ⟩

comparing the coefficients with⟨⟨T x, x

⟩⟩= ⟨⟨T x, x

⟩⟩= ∑α,β,γ∈On

eαeβ eγ⟨xα,Tβxγ⟩

yields (iii).


Remark 3.2.11In the second part of the last proof one could just as well have used a = i . The chosen approachhas the advantage that it works with non complex Clifford-Hilbert modules, i.e. Hilbert-Modulesof Rn over a Hilbert space, too.

EXAMPLE 3.2.2 (Multiples of the identity):We want to consider operators of the form a Id : Hn → Hn , x 7→ xa, where a ∈Cn . Proposi-tion 3.2.10 yields that these operators are self adjoint, iff

aα ∈R ∀α : eα = eα,

iR ∀α : eα =−eα.

Theorem 3.2.12LET T ∈ L(Hn) be self-adjoint.

THEN

‖T ‖ ≤ sup‖x‖≤1

|⟨⟨T x, x⟩⟩|.

Furthermore sup‖x‖≤1 |⟨⟨

T x, x⟩⟩| ≤ 2n/2‖T ‖.

Proof. Since(Hn , ( · , · )

)is a complex Hilbert space

‖T ‖ = sup‖x‖≤1

|(T x, x)| ≤ sup‖x‖≤1

|⟨⟨T x, x⟩⟩|.

(See for example [56] V.5.7.)

Using Theorem 3.2.4(iii) we know that

sup‖x‖≤1

|⟨⟨T x, x⟩⟩| ≤ 2n/2 sup

‖x‖≤1‖T x‖‖x‖ = ‖T ‖.

Corollary 3.2.13LET T ∈ L(Hn) be self-adjoint.

THEN ⟨⟨T x, x

⟩⟩= 0 ⇔ T = 0.

3.2.4 The Fourier transform on Lp (Rn)n

We extend the Fourier transform in a standard way to Clifford algebra valued functions.

Definition 3.2.14 (Fourier transform on Clifford-Hilbert modules)LET f =∑

j∈One j f j ∈ L1(Rn)n .

THEN the Fourier transform of f is defined as

F ( f )(ξ) = f (ξ) := ∑j∈On

e j f j (ξ) =∫Rn

f (x)e2πi ⟨x,ξ⟩d x.

This definition is then extended to the corresponding Lp -modules, 1 ≤ p <∞, in the usual way.


The Euclidean scalar product on L2(Rn)n is given by

( f , g ) := ∑i∈On

⟨ fi , gi ⟩L2(Rn ) =ℜ(∫Rn

f (x)g (x)d x).

This scalar product satisfies ‖ f ‖22 = ( f , f ).

Theorem 3.2.15 (The Plancherel equality)LET f , g ∈ L2(Rn)n .

THEN⟨⟨

f , g⟩⟩= ⟨⟨

f , g⟩⟩

. Consequently ( f , g ) = ( f , g ).

Proof. Let f =∑j∈On

e j f j , where f j ∈ L2(Rn ,C) ∀ j ∈Nn . Then

⟨⟨f , g

⟩⟩= ∑j ,k∈On

e j ek⟨ f j , gk⟩L2(Rn )(i )= ∑

j ,k∈On

e j ek⟨ f j , gk⟩ =⟨⟨

f , g⟩⟩

.

In (i ) we used the Plancherel inequality for L2(Rn ,C).

3.3 Clifford analysis

Our goal is to define an analytical signal inRn . In Theorem 2.1.2 and Theorem 2.1.7 we haveseen that the Riesz transforms are a suitable extension of the Hilbert transform to higher di-mensions. In this chapter we will introduce a hyper-complex analysis that is built on a set ofgeneralized Cauchy Riemann equations that can characterize the analytical signal in one andhigher dimensions as a boundary value of a function satisfying these equations.

This hyper-complex analysis is called Clifford analysis. We will only give a short introductionto Clifford analysis. For details see [13], [21] and [19].

For n ∈N we will consider the Clifford algebras Rn , and functions in C 1(Rn+1,Rn+1), whereRn+1 is identified with the real vector space of paravectors spaneαn

α=0 inRn . In this manner,we will not encounter any zero divisors.

Functions f :Rn+1 7→Rn+1 in this setting are of the form

f (x) =n∑α=0

fα(x)eα,

where fα :Rn+1 →R, α= 0, . . . ,n.

The Dirac operator is given by

D :=n∑β=1

∂

∂xβeβ

and the Cauchy-Riemann operator by

∂ :=n∑β=0

∂

∂xβeβ,

clearly ∂= ∂∂x0

+D .

Since multiplication in this algebra is not commutative, these operators may act from the leftor from the right.

3.3. CLIFFORD ANALYSIS 59

We will assume that a differential operator defined as d :=∑nβ=0

∂∂xβ

eβ usually acts from theleft by

d f (x) = ∑α∈On

∑β

∂ fα∂xβ

(x)eβeα.

The operator acting on a function from the right is then denoted by

( f d)(x) = ∑α∈On

n∑β=0

∂ fα∂xβ

(x)eαeβ.

Definition 3.3.1 (Monogenic Function)LET f :Rn+1 →Rn be a partially differentiable function such that f fulfills the Cauchy-Riemanntype equation

∂ f (x) = ∑α∈On

n∑β=0

∂ fα∂xβ

eβeα(x) = 0,

where f (x) =∑α∈On

eα fα(x).

THEN f is called (left-) monogenic.

A partially differentiable function f :Rn+1 →Rn fulfilling the Cauchy-Riemann-type equation( f ∂) = 0 is called right monogenic.

Theorem 3.3.2 (Left- and right monogenic functions)LET f :Rn+1 →Rn+1 be a left monogenic function.

THEN f is right monogenic.

Proof. Let f be a left monogenic function. Then

∂ f =n∑

α,β=0eαeβ

∂

∂αfβ =

n∑α=0

eαeα∂

∂αfα+

∑α∈1,...,n

eα( ∂

∂αf0 + ∂

∂0fα

)+ ∑α,β∈1,...,n

α<β

eαeβ( ∂

∂αfβ−

∂

∂βfα

)

= 0.

It follows that∂

∂0f0 −

∑α∈1,...,n

∂

∂αfα = 0,

( ∂

∂αf0 + ∂

∂0fα

)= 0, ∀α ∈ 1, . . . ,n

and∂

∂αfβ =− ∂

∂βfα, ∀α 6=β ∈ 1, . . . ,n.

As a consequence

0 =n∑α=0

eαeα∂

∂αfα+

∑α∈1,...,n

eα( ∂

∂αf0 + ∂

∂0fα

)+ ∑α,β∈1,...,n

α<β

eβeα( ∂

∂αfβ−

∂

∂βfα

)=

n∑α,β=0

eαeβ∂

∂αfβ

= ( f ∂).


Remark 3.3.3As a consequence of Theorem 3.3.2 we will henceforth simply speak of monogenic functions.

Theorem 3.3.2 does not state that ∂ f = ( f ∂)

Corollary 3.3.4Let f ∈ C 1(Rn+1,Rn+1). Then monogenicity is equivalent to a set of Cauchy-Riemann equa-tions for the component functionals fα :

∂ fα∂xβ

(x) = ∂ fβ∂xα

(x), ∀α,β= 1, . . . ,n

∂ fα∂x0

(x)+ ∂ f0

∂xα(x) = 0, ∀α= 1, . . . ,n (3.4)

n∑α=1

∂ fα∂xα

(x) = ∂ f0

∂x0(x).

Proof. We have proven this in the proof of Theorem 3.3.2.

Remark 3.3.5Notice that the Dirac operator D = ∑n

β=1∂∂xβ

eβ yields the Cauchy-Riemann like system (2.2).

That is let f =∑n+1α=1 eα fα ∈ L2(Rn ,Rn+1) ⊂ L2(Rn)n+1 be continuously differentiable. Then

D f = 0

iff

n+1∑α=1

∂ fα∂xα

= 0,

∂ fα∂xβ

= ∂ fβ∂xα

; α,β= 1, . . . ,n +1.

This is exactly (2.2) if we set u0 = fn+1 and uα = fα, ∀α= 1, . . . ,n.

EXAMPLE 3.3.1 (Monomials):Let 1 < n ∈ N. Then the monomials xd , x ∈ Rn+1; d ∈ N are not monogenic on the wholespace:

Let d = 1, that is x1 =∑nα=0 eαxα. Then

∂x = ∂x0

∂x0+

n∑α=1

eαeα∂xα∂xα

= 1−n.

Let d = 2, that is

x2 =(n∑α=0

eαxα)(n∑β=0

eβxβ)

=n∑α=0

e2αx2

α+n∑

α,β=0(eαeβ+eβeα)xαxβ

= x20 −

n∑α=1

x2α+2

n∑α=1

eαxαx0 +n∑

α,β=1(eαeβ−eαeβ)xαxβ

= x20 −

n∑α=1

x2α+2

n∑α=1

eαxαx0.


Then

∂x2 =∂(x20 +2

∑nα=1 eαxαx0)

∂x0+

n∑α=1

eα∂(2eαxαx0 −x2

α)

∂xα

= 2x0 +2n∑α=1

eαxα−2n∑α=1

eαxα−2n∑α=1

x0

= 2x0(1−n).

For the complex case, i.e. n = 1 and Rn = C, where 1−n = 0, the terms containing (1−n)vanish, and thus the monomials are analytical, as is well known.

EXAMPLE 3.3.2 (Fueter variables):A Fueter variable is defined as zα(x) := xα− eαx0. This function is clearly monogenic and, ifwe define z0 = x0 we gain a basis forRn+1 consisting of monogenic elements.

The product of two monogenic functions is not in general everywhere monogenic on Rn+1.Take for example the product of the Fueter variables

zαzβ(x) = xαxβ−eβx0xα−eαx0xβ+eαβx20 .

Then,

∂zαzβ(x) =∂(eαβx20 −eβx0xα−eαx0xβ)

∂x0+eα

∂(xαxβ−eβx0xα)

∂xα+eβ

∂(xαxβ−eαx0xβ)

∂xβ

= 2eαβx0 −eβxα−eαxβ+eαxβ−eαβx0 +eβxα+eαβx0

= 2eαβx0.

3.3.1 The exponential function

Theorem 3.3.6 (Paravector-valued Exponential Function)By

exp :Rn+1 →Rn+1, x 7→ ∑k∈N0

xk

k !

an exponential function for paravectors which is of the form

exp(x) = ex0(

cos |~x|+ ~x

|~x| sin |~x|), ∀x ∈Rn+1 (3.5)

is defined.

This function is not monogenic as follows from Example 3.3.1 but has the usual properties ofan exponential function if restricted to the subspaceR+~xR, which can be identified withCsince ~x

|~x| ∈ Sn . It holds true that

(i) exp(x + y) = exp(x)exp(y), ∀x y = y x, x, y ∈Rn+1;

(ii) exp(−x)exp(x) = 1, exp(x) 6= 0;

(iii) exp(kx) = (exp(x))k ;

(iv) ∂exp(x) = ex0 n−1|~x| sin(|~x|).


Proof. Let x = x0 +~x ∈Rn+1 and i := ~x|~x| . It follows that i 2 = −1. Note that the elements xk

k !of the series defining the exponential function are contained in the subalgebraR+ iR ⊂Rn ,which is isomorphic toC.

Hence writing x = x0 + i |~x| we know that

exp(x) = ex0(

cos(|~x|)+ i sin(|~x|))= ex0(

cos(|~x|)+ ~x

|~x| sin(|~x|)).

Now equations (ii) and (iii) follow from the corresponding properties of the exponential func-tion inC.

(i) is easy to compute, the proof is exactly the same as in the case of the exponential functionof matrices.

(iv)

∂exp(x) =n∑α=0

eα∂

∂xα

(ex0

(cos(|~x|)+

n∑β=1

eβxβ|~x| sin(|~x|)))

= ∂ex0 exp(~x)

∂x0+ex0

( n∑α=1

eα∂cos(|~x|)∂xα

+n∑β=1

eαeβ∂

∂xα

xβ|~x| sin(|~x|)

)= ex0

(cos(|~x|)+ ~x

|~x| sin(|~x|))+ex0(−

n∑α=1

eαxα sin(|~x|)

|~x| +n∑β=1

eαeβxβ|~x| cos(|~x|) xα

|~x|

+ sin(|~x|)( ∑β 6=α

eαeβ−xβxα

|~x|3 + x2α

|~x|3))

= ex0(

cos(|~x|)+ ~x

|~x| sin(|~x|)−n∑α=1

eαxα sin(|~x|)

|~x| − xα|~x| cos(|~x|) xα

|~x| + sin(|~x|) x2α

|~x|3)

= ex0n −1

|~x| sin(|~x|).

This holds true, since ∂∂xα

xα|~x| = ( 1

|~x| −x2α

|~x|3 ) = ∑β 6=α

x2β

|~x|3 and since the terms containing eαeβ arecancelled by the terms containing eβeα =−eαeβ.

Definition 3.3.7 (Phase of a Paravector)LET x = x0 +~x ∈Rn+1be a paravector.

THEN a phase φ, a phase direction d ∈ Rn and an amplitude a ∈ R+ are defined byx = a(cos(φ)+ d sin(φ)), whence a = |x|, d = ~x

|~x| and φ = arctan( x0|~x|

). (See Figure 3.1 for an

illustration.)

Remark 3.3.8 (Relation to spherical coordinates)In C the phase is closely related to the polar coordinates, since d = sgn(ℑ(x)) we have a de-composition into an amplitude, which corresponds to the radius, the phase angle and a phasesign. Phase angle and phase sign together correspond to the angle in polar coordinates: dφ isequal to the angle in polar coordinates. This leads to a continuous phase angleφ. The complexconjugate x of a x ∈C has the same phase angle φ as x, but the negative phase sign d .

For x ∈ Rn+1, xn := x = a(cos(φn)+dn sin(φn)) we can map the vector valued direction d n

to a paravector via Rn 7→ R(n−1)+1, dn = ∑ni=1 ei dn,i 7→ xn−1 = ∑n

i=1 ei−1dn,i and use the de-composition, where a = 1. This leads to a decomposition into one amplitude, n phase angelswith range [0,π[ and one sign d = d1. The phase angels φ2, . . . ,φn are equal to the angles of


e1 e2

e0

x

|x|

!x|!x|

!(x)

x1x2

x0

1

Figure 3.1: Phase φ and phase direction ~x‖~x‖ (direction of the dashed vector) of the 3-D vector

x = (x0, x1, x2).

the spherical coordinates that range between [0,π[. d1φ1 is then equal to the spherical angleranging between [0,2π[.

This yields a set of spherical coordinates forRn+1 of the form x = (x0, x1, . . . , xn), where

x0 = a cos(φn)

x1 = a sin(φn)cos(φn−1)

x2 = sin(φn)sin(φn−1)cos(φn−2)

......

xn−1 = ad1 sin(φn) · · ·sin(φ2)cos(φ1)

xn = ad1 sin(φn) · · ·sin(φ2)sin(φ1).

Explicitly in the case that n +1 = 3 this yields

x = a(

cos(φ),dθ sin(φ)cos(θ),dθ sin(φ)sin(θ)).

This differs slightly from the convention we use inR3. (See subsection A.1.2.)

Chapter 4

Monogenic signals

The agenda of this chapter is to formulate monogenic signals as an extension of analyticalsignals to higher dimensions and to examine the properties of monogenic signals.

The first step is the definition of a hypercomplex Riesz transform R, given in Definition 4.1.1,that extends the Hilbert transform iH . What is novel about this definition is that we definethe Riesz transform as a left linear-operator on a Clifford-Hilbert module. As a consequencethe (hypercomplex) Riesz transform we define is unitary and self-adjoint as is shown in Theo-rem 4.1.4. It follows that the Riesz transform is its own inverse. Furthermore, the Riesz trans-form is closely connected to differential operators as seen in a new result presented in Theo-rem 4.2.1.

We define monogenic signals in Definition 4.3.1, and in Definition 4.4.1 we state the decom-position of a monogenic signal into phase, amplitude, and phase direction. From the phasewe derive in Definition 4.4.2 an instantaneous frequency for the monogenic signal. From theproperties of the Riesz transforms it follows that a monogenic signal is generated by a projec-tion M : f 7→ fm (Theorem 4.3.3) that is closely connected to Clifford analysis. We state theseconnections in section 4.5.

The final section will show how a monogenic signal is uniquely defined as the extension ofan analytical signal by the condition that it is compatible with rotations, which generate thesymmetry group we have to deal with in image analysis.


4.1 Hypercomplex Riesz transforms

Depending on the kind of dilation we use, we need to define analytical signals in differentways. The key point here is to look at the invariances of the corresponding analog of the Hilberttransform.

The Riesz transform commutes with isotropic dilation and behaves like a vector under rota-tion. Thus, it is a tool of choice for a setting where rotations are important such as, for example,in image analysis.

Since we wish to be able to work with images and hence rotations we will now use the Riesztransform defined earlier to define a hyper-complex analytical signal. For this purpose, we

65

66 CHAPTER 4. MONOGENIC SIGNALS

introduce a hyper-complex Riesz transform, which plays a role similar to that of the Hilberttransform.

Definition 4.1.1 (Hypercomplex Riesz Transform)LET f ∈ Lp (Rn ,Rn), 1 < p <∞.

THEN we define the hypercomplex Riesz transform by

R :Lp (Rn ,Rn) → Lp (Rn ,Rn)

f 7→ R f =n∑α=1

Rα f eα =n∑α=1

∑β∈On

eβeαRα fβ.

Corollary 4.1.2 (Fourier multiplier of the hypercomplex Riesz transform)LET f ∈ Lp (Rn ,Rn).

THEN it follows from Theorem 2.1.2 (vii) that

F (R f )(ξ) =n∑α=1

∑β∈On

eβeαiξα‖ξ‖ fβ(ξ), f.a.a. ξ ∈Rn .

I.e. the Fourier multiplier of the hypercomplex Riesz transform is

R(ξ) =n∑α=1

eαiξα|ξ| , ∀ξ= (ξ1, . . . ,ξn) ∈Rn .

Corollary 4.1.3 (The hypercomplex Riesz transform of real valued functions)LET f be a real valued function f = e0 f0, f0 ∈ L2(Rn ,R).

THEN the Riesz transform maps f into the space of 1-vectors.

R :Lp (Rn ,R) → Lp (Rn ,Rn)

f 7→ R f =n∑α=1

eαRα f .

We will now proof that the Riesz transform is a self-adjoint, unitary and hence bijective opera-tor.

Theorem 4.1.4 (The hypercomplex Riesz transform is self-adjoint and unitary)Let f ∈ L2(Rn ,Cn). Then

(i) R is self-adjoint, i.e.⟨⟨

R f , g⟩⟩= ⟨⟨

f ,Rg⟩⟩

,

(ii) R is unitary, i.e.⟨⟨

R f ,Rg⟩⟩= ⟨⟨

f , g⟩⟩

and hence ‖R f ‖ = ‖ f ‖.

Proof. Let f , g ∈ L2(Rn ,Rn). Then

4.2. DIFFERENTIAL OPERATORS AND RIESZ TRANSFORMS 67

ad (i) ⟨⟨R f , g

⟩⟩= ∑α∈1,...,n

∑β,γ∈On

eβeαeγ

∫Rn

i xα|x| fβ(x)gγ(x)d x

= ∑α∈1,...,n

∑β,γ∈On

eβ(−eα)eγ

∫Rn

fβ(x)−i xα|x| gγ(x)d x

= ∑α∈1,...,n

∑β,γ∈On

eβeγeα

∫Rn

fβ(x)i xα|x| gγ(x)d x

= ⟨⟨f , Rg

⟩⟩.

ad (ii) ⟨⟨R f , Rg

⟩⟩=⟨⟨RR f , g⟩⟩= n∑

β,γ=1

∑α,δ∈On

eαeγeβeγ

∫Rn

−ξβξγ‖ξ‖2 fα(ξ)gδ(ξ)dξ

= ∑α,δ∈On

eα( ∑

1≤β<γ≤n(eγeβ−eβeγ)︸︷︷︸

=0

∫Rn

−ξβξγ‖ξ‖2 fα(ξ)gδ(ξ)dξ

+n∑β=1

eα e2β︸︷︷︸

=−1

eδ−ξ2

β

‖ξ‖2 fα(ξ)gδ(ξ))

=⟨⟨f , g

⟩⟩.

Now ‖R f ‖2 = (R f ,R f ) =ℜ(⟨⟨R f ,R f

⟩⟩)=ℜ(⟨⟨f , f

⟩⟩)= ( f , f ) = ‖ f ‖2.

4.2 Differential operators and Riesz transforms

The Riesz transform is closely connected to the Dirac and to the Laplace operator. The follow-ing equalities hold only for the hypercomplex Riesz transform on Hilbert-Clifford modules.

Theorem 4.2.1LET f ∈W 1(Rn).

THEN

(i) DR f = (−4)1/2 f ∈ L2(Rn).

(ii) ∂ f∂x j

= R j RD f ∈ L2(Rn).

Proof. ad(i) It is almost everywhere true that

F (DR f )(ξ) =n∑

j=12πe j iξ j

n∑k=1

i ekξk

|ξ| f (ξ)

= 2π( n∑

j=1

ξ2j

|ξ| −∑j<k

(e j k +ek j )(ξ jξk ))

f (ξ) = 2π|ξ| f (ξ)

=F((−4)1/2 f

)(ξ).


ad(ii) Again almost everywhere, we have that

F (R j RD f )(ξ) = iξ j

|ξ|n∑

k=1

ek iξk

|ξ|n∑

l=1(2πi )elξl f (ξ)

= 2πiξ j

( n∑k=1

ξ2k

|ξ|2 − ∑k<l

ξlξk

|ξ2| (ekl +el k ))

f (ξ) = 2πiξ j f (ξ)

=F( ∂ f

∂x j

)(ξ).

4.3 The monogenic signal

Now that we have decided on a Riesz transform as an extension of the Hilbert transform, wedefine monogenic signals as follows.

Definition 4.3.1 (Monogenic signal)LET f ∈ Lp (Rn)n = Lp (Rn ,Rn), where 1 < p <∞.

THEN the monogenic signal fm is defined by the following operator:

M : Lp (Rn)n → Lp (Rn)n ,

f 7→ fm = f +R f = f +n∑

j=1R j f e j .

In a similar form, this has been proposed by Felsberg in his PhD-thesis [15]. However, notethat the Riesz transform used to define monogenic signals in [15] is neither a left linear oper-ator nor unitary nor self adjoint, whence the resulting monogenic signal is not left-linear. Theleft-linearity of the monogenic signal, however, is necessary for the proof that the monogenicsignal maps frames of L2(Rn) to Clifford frames of a certain Clifford-Hilbert module, a clearlyfavorable property. (See section 5.3.)

Remark 4.3.2

(i) LET f ∈ L2(Rn) be a real valued signal.

THEN M e0 f = fm ∈ L2(Rn ,Rn+1) is a paravector valued function.

(ii) The interpretation of this monogenic signal as a local Fourier transform is not possible,as there is no Fourier transform that would fit: The monogenic signal is paravector val-ued, while the standard Fourier transform has values inCn . However, we can define aninstanteous frequency via the phase. (See section 4.4.)

Theorem 4.3.3 (The monogenic projection)The operator 2−1/2M is a projection.

Proof. Let f ∈ Lp (Rn)n . We have to show that M 2 = 2M :

M 2 f = (Id+R)(Id+R) f = (Id+2R + Id) f = 2M f .

4.4. THE PHASE OF A MONOGENIC SIGNAL 69

4.4 The phase of a monogenic signal

Our goal in defining a monogenic signal was to attain a decomposition of a signal into phaseand amplitude. This decomposition will be the topic of the present section.

Since the monogenic signal is paravector-valued, we use Definition 3.3.7 to decompose fm intoan amplitude function a, a phase φ, and a vector-valued phase direction d .

e1 e2

e0

fm(x)

|fm(x)|

Rf(x)|Rf(x)|

!(x)

R1f(x) R2f(x)

f(x)

1

Figure 4.1: Phase φ(x) and phase direction d(x) = R f (x)|R f (x)| (direction of the dashed vector) of the

3-D vector fm(x) = ( f (x),R1 f (x),R2 f (x)).

Definition 4.4.1 (Amplitude-phase decomposition of the monogenic signal)LET f ∈ Lp (Rn) and let fm ∈ Lp (Rn ,Rn+1) be its associated monogenic signal.

THEN a monogenic signal can be decomposed as follows.

fm = | fm |(

f

| fm | +R f

|R f ||R f || fm |

)= a

(cos

(φ

)+d sin(φ

))= a exp(dφ

). (4.1)

Here, a = | fm | ∈R+0 is the amplitude , φ= arccos

(f

|R f |)∈ [0,π[ is the phase, and d =

−→fm

|−→fm |= R f

|R f |is the vector-valued phase direction.

Definition 4.4.2 (Instantaneous frequency)LET f ∈ Lp (Rn) have the property that the phase φ is a differentiable function.

THEN the instantaneous frequencyω(x) is defined as the directional derivative in the directiond(x) of the phase angle φ(x) at any point x ∈Rn for which f (x) 6= 0:

∇φ(x) =(dφ(x)

d x1, . . . ,

dφ(x)

d xn

).


Theorem 4.4.3 (Computation of the instantaneous frequency)LET f be as in Definition 4.4.2.

THEN

ω(x) =⟨∇φ(x),d(x)

⟩Rn

=⟨ |R f (x)|∇ f (x)− (∇|R f (x)|) f (x)

a2(x),d(x)

⟩Rn

.

Proof. Since tan(φ(x)) = |R f (x)|f (x) , for all α ∈ 1, . . . ,n, x ∈Rn , and f (x) 6= 0 we have that

dφ(x)

d xα= d

d xαarctan

( f (x)

|R f (x)|)

= 1

1+(

f (x)|R f (x)|

)2

f (x) dd xα

|R f (x)|− |R f (x)| dd xα

f (x)

f 2(x)

=f (x) d

d xα|R f (x)|− |R f (x)| d

d xαf (x)

f 2(x)+ (R f (x)

)2 .

Hence,

ω(x) =⟨∇φ(x),d(x)

⟩Rn

=⟨ |R f (x)|∇ f (x)− (∇|R f (x)|) f (x)

a2(x),d(x)

⟩Rn

.

4.5 The monogenic signal and Cauchy transforms

Theorem 4.3.3 states that a monogenic signal is the scalar multiple of a projection. In thissection we determine the space that is the range of this projection.

This yields a relation between monogenic signals and some topics in (hyper-) complex anal-ysis. The results presented here are not new but they are important in obtaining a completeview of the monogenic signal.

We start by deriving a relation between monogenic signals, Hardy spaces, and the Cauchy,Poisson, and Riesz transforms.

Definition 4.5.1 (Hardy space)LET p > 0.

THEN the Hardy spaceH p (Rn+1

+ ) = H p (Rn+1+ ,Rn+1)

is the space of functions u :Rn+1+ →Rn+1, that are monogenic in the upper half plane

Rn+1+ := (x0, x) : x0 ∈R+, x ∈Rn,

and satisfy

‖u‖H p = supx0>0

(∫Rn

|u(x0, x)|p d x)1/p <∞.

Theorem 4.5.2 (Monogenic signals and conjugate harmonic functions)LET f =∑n

α=0 eα fα ∈ Lp (Rn ,Rn+1), 1 < p <∞, and let

uα(x, x0) := Px0 ∗ fα(x), ∀α= 0, . . . ,n,

4.5. MONOGENIC SIGNALS AND CAUCHY TRANSFORMS 71

where Px0 is the Poisson kernel defined in Definition 2.1.5 on page 23:

Px0 (x) := cnx0

(|x|2 +x20)

n+12

,

where

cn = Γ( n+12 )

πn+12

, l = 1, . . . ,n.

THEN

fα = Rα( f0), ∀α= 1, . . . ,n,

if and only if u ∈ H p (Rn+1+ ). That is, u is a monogenic function in the upper half-space andthus satisfies the generalized Cauchy Riemann equations (3.4)

∂uα∂xβ

(x, x0) = ∂uβ∂xα

(x, x0), ∀α,β= 1, . . . ,n,

∂uα∂x0

(x, x0)+ ∂u0

∂xα(x, x0) = 0, ∀α= 1, . . . ,n,

n∑α=1

∂uα∂xα

(x, x0) = ∂u0

∂x0(x, x0), ∀(x, x0) ∈Rn+1

+ .

Proof. Suppose that fα = Rα f0. Then fα(t ) = i tα|t | f0(t ) and, hence,

uα(x, x0) =∫Rn

f0(t )i tα|t | e−2π|t |x0 e2πi t x d t , ∀α= 1, . . . ,n,

u0(x, x0) =∫Rn

f0(t )e−2π|t |x0 e2πi t x d t .

By the dominated convergence theorem [41], we may differentiate under the integral sign, andthus obtain

∂u0

∂x0(x, x0) =−2π

∫Rn

f0(t )|t |e−2π|t |x0 e2πi t x d t ,

∂u0

∂xβ(x, x0) = 2πi

∫Rn

f0(t )tβe−2π|t |x0 e2πi t x d t ,

∂uβ∂x0

(x, x0) = 2πi∫Rn

f0(t )tαe−2π|t |x0 e2πi t x d t ,

∂uα∂xβ

(x, x0) = 2π∫Rn

f0(t )tαtβ|t | e−2π|t |x0 e2πi t x d t .

Now (3.4) is easy to check. The first equation follows from

∂uα∂xβ

(x, x0) =−2π∫Rn

f (t )tαtβ|t | e−2π|t |x0 e2πi t x d t

= ∂uβ∂xα

(x, x0),

and the second identity from

∂u0

∂xβ(x, x0) = 2πi

∫Rn

f (t )tβe−2π|t |x0 e2πi t x d t

=−∂uβ∂x0

(x, x0).


Finally, the last identity follows from

n∑α=1

∂uα∂xα

(x, x0) =−n∑α=1

2π∫Rn

f (t )t 2α

|t |e−2π|t |x0 e2πi t x d t

−= 2π∫Rn

f (t )

∑nα=1 t 2

α

|t | e−2π|t |x0 e2πi t x d t

=−2π∫Rn

f (t )|t |e−2π|t |x0 e2πi t x d t

= ∂u

∂x0(x, x0).

Conversely, letβ ∈ 1, . . . ,n and uβ(x, x0) = ∫Rn fβ(t )e−2π|t |x0 e2πi t x d t . The fact that ∂u0

∂xβ=− ∂uβ

∂x0,

implies that

2πi∫Rn

f (t )tβe−2π|t |x0 e2πi t x d t = 2π∫Rn

fβ(t )|t |e−2π|t |x0 e2πi t x d t ;

Therefore fβ(t ) = i tβ|t | f (t ), and thus

fβ = Rβ( f ), β= 1, . . . ,n.

Remark 4.5.3Let 1 < p <∞. We have a one to one correspondence between functions in Lp , monogenic sig-nals and functions on Hardy spaces in the upper half-space. As we have seen in Theorem 4.5.2,the Poisson transform of a monogenic signal, which is given by convolution with the Poissonkernel, is a monogenic function in the upper half-spaceRn+1.

Let 1 < p <∞. For every f ∈ Lp the function u :Rn+1+ →Rn , (x0, x) 7→ Px0 fm(x) is an elementof H p (Rn+1+ ).

In the case p = 1, the additional condition R f ∈ L1(Rn ,Rn) yields u ∈ H 1(Rn+1+ ,Rn+1).

Using the conjugate Poisson kernels

ql (x, x0) := cnxl

(|x|2 +x20)

n+12

= Px0 Rl (x),

we can write u in the form

u(x, x0) := u0(x, x0)+∑l

el ul (x, x0) := e0 f ∗Px0 (x)+n∑

l=1el f ∗ql (x, x0).

The monogenic function equals the limit

limx0→0

u(x0, x) = fm(x), a.e..

Furthermore, limx0→0∫Rn |u(x0, x)− f (x)|p d x = 0. (A proof for this can be found in [45], Chap-

terIII.2, Theorem 1.)

We have shown that the monogenic signal of a function in Lp is mapped by the Poisson trans-form to a function in the Hardy space H p . The next theorem gives a result in the oppositedirection, namely when a function in H p has a boundary value that is a function in Lp . Tostate this theorem, we need the notion of non-tangential limits.

4.5. MONOGENIC SIGNALS AND CAUCHY TRANSFORMS 73

Definition 4.5.4 (Non-tangential limit)LET c > 0. The cone based at x ∈Rn with aperture c is given by

Γc (x) := (y0, y) ∈Rn+1+ : |y −x| < cx0.

A function F onRn+1+ has a non-tangential limit at x, if

lim-n.t.z→x

F (z) = limz→x

z∈Γc (x)

F (z),

exists for some c > 0.

Theorem 4.5.5LET p > n−2

n−1 and u ∈ H p (Rn+1+ ).

THEN there exists an f ∈ Lp (Rn) such that

1. lim-n.t.z→x u(z) = f (x), a.e.;

2. limx0→0∫Rn |u(x0, x)− f (x)|p d x = 0.

Proof. See [19], chapter 2, section 5. (Note the different convention with respect to the sign ofthe Riesz transform used there.)

What remains to be shown is that the limit of the function u is again a monogenic signal. Thisis proved in Theorem 4.5.2.

However, this result also be derived using the Cauchy integral of Clifford analysis.

Definition 4.5.6 (Cauchy Integral)LET 1 ≤ p <∞, and f ∈ Lp (Rn ,Rn+1). Let z ∈Rn+1 \Rn , i.e., z is an element of the upper orlower half-space; specifically ⟨z⟩0 6= 0.

THEN the Cauchy integral is defined by

C f (z) := 1

cn

∫Rn

u − z

|u − z|n n(u) f (u)du,

where n(x) =−e0 is the outward pointing normal vector toRn+1+ which is constant.

Note that C f is monogenic on Rn+1+ . Furthermore the Cauchy integral is closely related tomonogenic signals: C f (x0, x) = 1

2

(Px0 ∗ f (x)+∑n

l=1 el ql (x0, · )∗ f (x))= 1

2 Px0 fm(x), ∀x0 > 0.

Theorem 4.5.7 (Cauchy integral and monogenic signals)LET either 1 < p <∞ and f ∈ Lp (Rn ,Rn+1), or p = 1 and f ∈ L1(Rn ,Rn+1),R f ∈ L1(Rn ,Rn+1).

THEN C f ∈ H p (Rn+1+ ,Rn+1) and

lim-n.t.z→x

C f (z) = 1

2fm(x), a.e..

Proof. See [19], chapter 2, section 5.


Theorem 4.5.8LET 1 ≤ p <∞ and u ∈ H p (Rn+1+ ,Rn+1). Denote µ := lim-n.t.z→· u(z).

THEN

u =Cµ.

Proof. See [19], chapter 2, section 5.

Remark 4.5.9 (The Cauchy integral in the lower half-space)There is an analogous theory for Hardy spaces of the lower half-space. Let f ∈ Lp (Rn ,Rn+1),where 1 ≤ p <∞. Let z ∈Rn+1 \Rn , i.e., z is an element of the lower half-space; specifically⟨z⟩0 6= 0 and let n(u) = e0 be the outward pointing normal vector of Rn ⊂ Rn+1. If we letH p (Rn+1− ,Rn+1) be the Hardy space of the lower half-space, then the Cauchy integral

C f (z) := 1

cn

∫Rn

z −u

|u − z|n n(u) f (u)du = 1

2

(Px0 ∗ f (x)−

n∑l=1

el ql (x0, · )∗ f (x))

has the following properties.

Let either 1 < p <∞ and f ∈ Lp (Rn ,Rn+1), or p = 1 and f ,R f ∈ Lp (Rn ,Rn+1).

Then C f ∈ H p (Rn+1− ,Rn+1), and

lim-n.t.z→x

C f (z) = 1

2f (x)−R f (x), a.e..

Let 1 ≤ p <∞ and u ∈ H p (Rn+1− ,Rn+1).

Thenu =C (lim-n.t.

z→· u(z)).

Remark 4.5.10 (Range of a monogenic signal)The non tangential limits of Cauchy integrals define singular operators. These operators

P± : L2(Rn) → L2(Rn), f 7→ 1

2( f ±R f )

are projections called Plemelj projections. They satisfy

P+P− = P−P+ = 0, P 2± = P±, and P++P− = Id.

Furthermore, the Cauchy integral is the Poisson transform of the corresponding Plemelj pro-jection.

Let 1 < p <∞, thenT H p

± (Rn) := f ±R f : f ∈ Lp (Rn ,Rn+1),

and, for p = 1,T H 1

±(Rn) := f ±R f : f ,R f ∈ L1(Rn ,Rn+1).

P± are obviously projections onto T H p± (Rn). Furthermore,

M f = fm = 2P+ f , ∀ f ∈ T H+(Rn).

Let 1 ≤ p <∞. Then the Poisson transform is an isomorphism between H p (Rn+1± ,Rn+1) and

T H p± (Rn). Moreover, for 1 < p <∞, the operator

M± : Lp (Rn ,R) → T H p± (Rn), f 7→ ( f ±R f )

4.6. ALTERNATIVE ANALYTICAL SIGNALS 75

is an isomorphism with inverse

M−1± : T H p

± (Rn) → Lp (Rn ,R), f 7→ℜ f .

Let 1 ≤ p <∞. Then the Cauchy integral is an isomorphism between Lp (Rn) and H p (Rn+1± ,Rn+1).

In addition, the Cauchy integral is an isomorphism between T H p (Rn) and H p (Rn+1± ,Rn+1).

To summarize, we give a list of the operators used in this section:

monogenic signal M : Lp (Rn) →T H p+ (Rn), f 7→ fm ;

real part ℜ : T H p+ (Rn) →Lp (Rn), fm 7→ f ;

Poisson transform P : T H p+ (Rn) →H p (Rn+1

+ ,Rn+1), fm 7→C f ;

Cauchy transform C : Lp (Rn) →H p (Rn+1+ ,Rn+1), f 7→C f ;

tangential limes lim-n.t. : H p (Rn+1+ ,Rn+1) →T H p

+ (Rn), C f 7→ fm .

4.6 Alternative analytical signals

An analytical signal is uniquely defined in two different ways: On the one hand, the Poissontransform of an analytical signal satisfies the Cauchy Riemann equations as stated in Theo-rem 4.5.2. On the other hand, an analytical signal is defined by the Hilbert transform, which isis uniquely defined modulo a multiplicative constant by the following four properties stated inTheorem 1.4.2. The Hilbert transform

(i) anti-commutes with reflection, i.e. H ( f (− · ))(x) =−H ( f )(−x);

(ii) commutes with translation;

(iii) commutes with dilation;

(iv) ∃A(p) such that ‖H f ‖p ≤ A(p)‖ f ‖p ∀ f ∈ Lp (R).

Properties (ii) and (iv) are extended to higher dimensions in the obvious way. There are how-ever different approaches to extend properties (i) and (iii) to higher dimensions. Each of theseapproaches yields a uniquely defined extension of the Hilbert transform. The important prop-erty here is property (i), which states the behaviour of the Hilbert transform under a group ofsymmetries, i.e. the reflections. Dilation is then determined by the condition that it commuteswith these symmetries.

Remark 4.6.1 (Uniqueness of the Riesz transform)For the definition of the Riesz transform we chose as symmetries the group of reflections androtations inRn given by the matrix group O(n). As a consequence we extend property (iii) byisotropic dilations and property (i) by the corresponding property with respect to rotations andreflections inRn stated in Theorem 2.1.2(i). This yields the steerability of the Riesz transformDefinition 2.1.9. Together with properties (ii) and (iv) the Riesz transform is uniquely defined.

The identity of objects in images is not changed by isotropic dilations and rotations. Hence forimage analysis it is important that the action of rotation on images can be controlled. This isjust what steerability of the Riesz transform means. Property (i) states that Riesz transforms area representation of the rotation group. That is, the Riesz transform originates from symmetrieswhich are exactly those needed for image processing.


There are other representations of the rotation group on which the higher Riesz transforms inchapter 8 are based.

We have seen in Theorem 4.5.2 that the Poisson transform of hypercomplex monogenic signalssatisfies the Cauchy-Riemann equations as stated in Theorem 4.5.2. This is another aspectunder which the monogenic signal is a unique extension of the analytical signal.

The Riesz transform is not invariant under dilations of the form

Da1,...,an = diag(a1, . . . , an). (4.2)

Thus a different transform is needed to define an analytical signal in this setting. The partialHilbert transform with axis xα, α= 1, . . . ,n, is given by the Fourier multiplier

Hα := i xα|xα|

.

Partial Hilbert transforms are invariant under dilations of the form (4.2). Hence the transformof choice is the combination of all partial Hilbert transforms:

H f = ∑α∈On

eαHα f , eα ∈Rn ,

whereα= (α1, . . . ,α|α|) is a multiindex and Hα =Hα1 . . .Hα|α| . However, these partial Hilberttransforms do not commute with rotations. The correspondence with property (i) is the factthat partial Hilbert transforms anti-commute with reflections in the plane perpendicular tothe axis of the partial Hilbert transform, and commute with all reflections in planes that con-tain the axis of the partial Hilbert transform. In the two dimensional case this is the Hilberttransform proposed by Bülow in his thesis [8] and Baraniuck in [9].

This definition of an multidimensional analytical signal can be interpreted as an analyticalsignal where the term analytical corresponds to a different kind of Cauchy-Riemann like equa-tions.

InRn ×R+, take the Poisson kernel p(x, y) = yπ‖x+y‖2 and its conjugate q(x, y) = x

π‖x+y‖2 .

The analytical function of a 2-D function corresponding to the Quaternion Wavelet Transformof Baraniuck [9] found in the PhD-thesis of Bülow´s [8] would be u + i vi + j v j +kvk , where

u := f ∗p(x1, y1)∗p(x2, y2);

vi := f ∗q(x1, y1)∗p(x2, y2);

v j := f ∗p(x1, y1)∗q(x2, y2);

vk := f ∗q(x1, y1)∗q(x2, y2).

Thus, the analytical extension can be written as

(u + i vi )+ j (v j + i vk ) := u2 + j v2 = u1 + i v1 =: (u + j v j )+ i (vi + j vk ),

satisfying the Cauchy-Riemann-like equations given in [34]:

∂ul

∂yl+ ∂vl

∂xl= 0,

∂vl

∂yl+ ∂ul

∂xl= 0, l = i , j .

Chapter 5

Monogenic wavelets

The agenda of this chapter is to implement the monogenic signal defined in the last chaptervia wavelet frames.

The first section 5.1 motivates the use of wavelet frames for the implementation. Example 5.1.2illustrates our approach using the example of analytical wavelet frames and sets the roadmapfor sections 2 and 3.

Section 5.2 extends the concept of frames on Hilbert spaces to that of Clifford-Hilbert modules– the space in which the monogenic signal lives – introduced in section 3.2. Our concept ofClifford frames – which use Clifford algebra valued frame coefficients to achieve a frame de-composition – appears to be entirely novel and it is especially remarkable that Theorem 5.2.3holds although the Cauchy-Schwartz inequality is not valid for the Clifford algebra valued in-ner product

⟨⟨ · , · ⟩⟩.

Section 5.3 defines the notion of a hypercomplex wavelet transform, which is compatible withClifford frames (Definition 5.3.1) and shows that we can derive a monogenic wavelet framefrom a wavelet frame for L2(Rn) (Theorem 5.3.3). Furthermore, Theorem 5.4.3 states that thedecay rate of the wavelet frames is preserved by the Riesz transform.

In section 5.4 we derive novel conditions under which the partial Riesz transform of a functioninherits the decay rate.

Section 5.5 is dedicated to the search for suitable wavelet frames. First, we introduce resultswhich reduce the problem of constructing frames to the problem of constructing Riesz par-titions of unity. Theorem 5.5.7 provides a new way to find such Riesz partitions of unity. Anexplicit construction is then given in Example 5.5.1. Theorem 5.5.8 shows how to derive Rieszpartitions of unity from given compactly supported wavelet orthonormal bases of L2(R).

Finally section 5.6 gives the implementation of the wavelet frames of Example 5.5.1 as an im-ageJ plugin. The software and the applications have been the topic of the diploma thesis ofMartin Storath [47] under supervision of the author.


77

78 CHAPTER 5. MONOGENIC WAVELETS

5.1 Introduction and example

In the last chapter we showed that monogenic signals based on the Riesz transform are thenatural extension of analytical signals to arbitrary finite dimension in the context of imageprocessing.

Hence it is time to think about the implementation of the monogenic signal. The way we wantto do this is by using monogenic wavelets. There are three main reasons to use wavelets here:

The first reason to use wavelets is that the multi-scale phase and amplitude of the wavelets aremore useful than the phase of the monogenic signal, which can be interpreted as the averageof the phase over all scales. To demonstrate this let us consider an example.

EXAMPLE 5.1.1 (Phase separation via wavelets):LET

f (t ) = a(t )cos(ωt ),

where a ∈ L2(R) : supp(a) ∈]−ω,ω[.

THEN by the Bedrosian identity Theorem 2.2.16 on page 36 its analytical signal is a(t )e iωt . Theamplitude of this function is a(t ), its phase is ωt , and the instantaneous frequency is ω.

LET us now consider the function

g (t ) = a(t )(cos(ωt )+cos(γt )),

where 0 <ω< γ<∞.

THEN the amplitude of g is a(t )|cos(ω−γ2 t )|, its phase is ω+γ2 t and the instantaneous frequency

is ω+γ2 . That is the phase is an average of the two phases which are present. The amplitude

clearly has some component which we would consider to be part of the phase.

LET 2 j Tkψ j ,k∈Z be a wavelet set such that supp(ψ) ⊆ [−1/4,−1/8]∪ [1/8,1/4] and let φ ≥ ω.By Theorem 1.3.15 on page 10 we know that the Fourier transform of a(t )cos(φt ) is supportedonφ+supp(a). Let d := sup

|x−y | : x, y ∈ supp(a), aφ := blog2(φ−d)c and bφ := dlog2(φ+d)e.

THEN it follows that supp(F

(a cos(φ · )

)) ⊆ [2aφ ,2bφ ] and if bω < aγ then the wavelet trans-

form separates the two components which may then be analysed separately as shown in Ex-ample 5.1.2. Since the analytical wavelet coefficients of a cos(φ) have disjoint support withrespect to scale from those of a cos(ω) the phase of these two signals is well separated by theanalytical wavelet.

The second reason to use wavelets for the implementation of the Riesz transform is that theFourier multiplier of the Riesz transform is discontinuous at 0. Thus the Riesz transform ofa signal in L1(Rn) is not an element of L1(Rn) if its 0-th moment does not vanish. In fact,fast decay of the Riesz transform of a signal requires vanishing moments as shown in Theo-rem 5.4.3. As a consequence we will use wavelets with vanishing moments to implement theRiesz transform.

The third reason is the fact that the Riesz transform behaves remarkably well under the opera-tors which generate the wavelet system from the mother wavelet. Indeed, the Riesz transformis uniquely defined by the properties that it commutes with translation and isotropic dilationand behaves like a vector under rotation (Theorem 2.1.2 on page 20). That is Theorem 2.1.2(i)yields an irreducible unitary representation of the rotation group. Hence the dual frame of theRiesz transformed wavelet frame is the Riesz transform of the dual wavelet frame as shown inTheorem 5.3.4.

5.1. INTRODUCTION AND EXAMPLE 79

Signal f Monogenic Signal fmDecomposition into

Amplitude and Phase

Multiscale Detailsof f

Multiscale Detailsof the Monogenic Signal

Multiscale Details ofAmplitude and Phase

R

R

W WW ! R = R ! W

Figure 5.1: Commutative diagram for the Riesz (R) and wavelet (W ) transform. For a multiscaledecomposition of the details of amplitude and phase we can use either R W or W R, sincethey are equal.

The example of analytical wavelets in the 1−D case will demonstrate how the implementationvia wavelets works.

EXAMPLE 5.1.2 (Analytical wavelets):LET ψ ∈ L2(R) be a mother wavelet for a wavelet frame.

THEN Hψ, the Hilbert transform of ψ, is a mother wavelet that generates a wavelet frame forL2(R,C). Furthermore,ψa the analytical signal ofψ, is a mother wavelet for a wavelet frame ofT H 2−(R) – the space of tangential limits of functions in the Hardy space – with complex framecoefficients.

LET f ∈ L2(R) and let fa be its analytical signal.

THEN

Wψa f =Wψ fa

and hence we can compute the analytical signal of f via

fa =W −1ψ Wψa f .

Proof. The Hilbert transform is a bounded invertible mapping in L2(R). It follows by Theo-rem 1.3.27 on page 13, that Hψ generates a frame for L2(R).

We know that the operator mapping a signal to an analytical signal to be bounded and surjec-tive onto T H 2−(R). Hence Theorem 1.3.27 shows that ψa generates a frame for T H 2−(R).

The wavelet transform of f is

Wψa f (d , t ) = ⟨Dd Ttψa , f ⟩= ⟨Dd Tt (ψ+ iHψ), f ⟩= ⟨Dd Ttψ, f ⟩+ i ⟨H Dd Ttψ, f ⟩= ⟨Dd Ttψ, f ⟩+

∫R

sgn(ξ)Dd Ttψ(ξ) f (ξ)dξ

= ⟨Dd Ttψ, f ⟩− i ⟨Dd Ttψ,H f ⟩=Wψ fa(d , t ).

Here we used the fact that the Hilbert transform commutes with translation and dilation The-orem 1.4.2 on page 15 and Parsevals formula Corollary 1.3.14 on page 10.


5.2 Frames on Clifford-Hilbert modules

We wish to extend the analytical wavelets of Example 5.1.2 to our monogenic signal Defini-tion 4.3.1. Hence we need to define wavelet frames on Clifford-Hilbert modules. That is weneed a concept of frames in Clifford-Hilbert modules with frame coefficients in the Cliffordalgebra. These frames will be called Clifford frames. They will be generated using an analy-sis operator which is based on the Clifford algebra valued product

⟨⟨ · , · ⟩⟩rather than on the

scalar product( · , · )

. (See subsection 3.2.2 for the definitions of⟨⟨ · , · ⟩⟩

and( · , · )

.)

In the following we follow the general procedure for Hilbert spaces in [10], Chapters 3.2 and 5,modify and construct new proofs for the case of Clifford-Hilbert modules when necessary. Itis possible to roughly follow the proofs of the Hilbert space case because the Hilbert module(Hn ,

⟨⟨·, ·⟩⟩) over the Hilbert space H is in itself a Hilbert space(Hn , (·, ·)).

Difficulties in adapting the proofs are due to three facts:

• The coefficients of the Clifford frame are not scalars with respect to the Hilbert space(Hn , (·, ·)).

•⟨⟨ · , · ⟩⟩

is not a scalar product.

• The Clifford algebra setting we consider is not commutative.

Considering the modifications in the Cauchy Schwartz inequality Theorem 3.2.4(iii) on page50 it is especially remarkable that Theorem 5.2.3 holds with the same bounds as for Besselsequences in Hilbert spaces.

An outstanding feature of the Clifford frames is that they give a unified framework for framesand multi-frames. (See Theorem 5.3.3 for an example.)

In the following(H ,⟨ · , · ⟩) will be a Hilbert space and

(Hn ,

⟨⟨ · , · ⟩⟩)will be the Clifford-

Hilbert module defined in subsection 3.2.2.

5.2.1 Bessel sequences

Lemma 5.2.1 (The Synthesis and the Analysis Operator)LET fk k∈N ⊂ Hn be a sequence in the Clifford-Hilbert module Hn , and suppose that

∑k∈N ck fk

is convergent for all ck k∈N ∈ l 2n(N) ∼= l 2(N,Cn).

THEN the synthesis operator

T : l 2n(N) 7→ Hn , ck 7→ ∑

k∈Nck fk

defines a boundedCn-left-linear operator. The adjoint, called the analysis operator, is givenby

T ∗ : Hn 7→ l 2n(N), f 7→

⟨⟨f , fk

⟩⟩k∈N.

Furthermore ∑k∈N

∣∣⟨⟨ f , fk⟩⟩∣∣2 ≤ ‖T ‖2‖ f ‖2, ∀ f ∈ Hn .

5.2. FRAMES ON CLIFFORD-HILBERT MODULES 81

Proof. Let l ∈N. Consider the sequence of bounded left-linear operators

Tl : l 2n(N) 7→ Hn , ck k∈N 7→

l∑k=1

ck fk .

Clearly Tl → T pointwise as l → ∞, thus T is bounded. (See [56] IV.2.5, a corollary of theBanach-Steinhaus theorem.)

Now let f ∈ Hn , ck k∈N ∈ l 2n(N). Then

⟨⟨f ,T (ck k∈N)

⟩⟩Hn

= ⟨⟨f ,

∑k∈N

ck fk⟩⟩= ∑

k∈N

⟨⟨f , fk

⟩⟩ck

= ∑k∈N

∑α,β∈On

eαeβ⟨⟨⟨

f , fk⟩⟩⟩α⟨ck⟩β = ⟨⟨

⟨⟨

f , fk⟩⟩

k∈N, ck k∈N⟩⟩

l 2n (N). (5.1)

Since T is bounded, T ∗ is a bounded linear operator T ∗ : Hn → l 2n(N). Hence the k-th co-

ordinate functional is bounded from Hn 7→Cn . By the Riesz representation theorem (Theo-rem 3.2.5 on page 52) there exists a sequence gk ⊂ Hn such that

T ∗ f = ⟨⟨f , gk

⟩⟩Hn

k∈N.

By (5.1) it follows that

∑k∈N

⟨⟨f , fk

⟩⟩Hn

ck = ⟨⟨f ,T (ck k∈N)

⟩⟩Hn

= ⟨⟨T ∗ f , ck k∈N

⟩⟩l 2

n (N) =∑

k∈N

⟨⟨f , gk

⟩⟩Hn

ck ,

and hence gk = fk .

Now∑

k∈N |⟨⟨ f , fk⟩⟩|2 = ‖T ∗ f ‖2 ≤ ‖T ∗‖2‖ f ‖2 = ‖T ‖2‖ f ‖2.

Definition 5.2.2 (Bessel Sequence)LET fk k∈N ⊂ Hn be a sequence such that there exists a constant B > 0 satisfying

∑k∈N

|⟨⟨ f , fk⟩⟩|2 ≤ B‖ f ‖2 ∀ f ∈ Hn .

THEN fk k∈N is called a Bessel sequence. A number B satisfying this inequality is called aBessel bound.

Theorem 5.2.3 (Boundedness of the synthesis operatr)LET fk k∈N ⊂ Hn be such that the synthesis operator

T : l 2n(N) → Hn , ck k∈N 7→ ∑

k∈Nck fk

is a well defined left-linear bounded operator and ‖T ‖ ≤pB .

THEN AND ONLY THEN is fk k∈N a Bessel sequence with Bessel bound B .


Proof. Let fk k∈N ⊂ Hn be a Bessel sequence with Bessel bound B .

We will first show that∑

k∈N ck fk is convergent. Let 0 < l < m <∞.∥∥∥ m∑k=1

ck fk −l∑

k=1ck fk

∥∥∥=∥∥∥ m∑

k=l+1ck fk

∥∥∥ 3.2.4(vi i )≤ sup‖g‖≤1g∈Hn

∣∣∣( m∑k=l+1

ck fk , g)∣∣∣

≤ sup‖g‖≤1

m∑k=l+1

|(ck fk , g )| = sup‖g‖≤1

m∑k=l+1

∣∣ℜ(ck

⟨⟨fk , g

⟩⟩)∣∣= sup

‖g‖≤1

m∑k=l+1

∣∣∣ ∑α∈On

|eα|2⟨ck⟩α⟨⟨⟨

fk , g⟩⟩⟩

α

∣∣∣∗≤ sup

‖g‖≤1

m∑k=l+1

( ∑α∈On

∣∣⟨ck⟩α∣∣2

)1/2( ∑α∈On

∣∣⟨⟨⟨ fk , g⟩⟩⟩α∣∣2

)1/2

= sup‖g‖≤1

m∑k=l+1

|ck |∣∣⟨⟨ fk , g

⟩⟩∣∣≤ ( m∑k=l+1

|ck |2)1/2

sup‖g‖≤1

( m∑k=l+1

|⟨⟨ fk , g⟩⟩|2)1/2

≤p

B( m∑

k=l+1|ck |2

)1/2

For (*) we used the Cauchy-Schwartz inequality in the complex Hilbert space(Cn , (·, ·)).

Since ck k∈Z ∈ l 2n(N) this shows that

∑mk=1 ck fk

∞m=1 is a Cauchy sequence in Hn and hence

convergent as a sequence in the complex Hilbert space(Hn , (·, ·)). Thus T (ck k∈N) is well de-

fined. T is obviously left-linear. Choosing m = 0 and letting l → ∞ in the calculation abovegives the boundedness of T .

The second statement has already been shown in Lemma 5.2.1.

Corollary 5.2.4LET fk k∈N ⊂ Hn be a sequence and let

∑k∈N ck fk be convergent for all ck k∈N ∈ l 2

n(N).

THEN fk k∈N is a Bessel sequence.

Corollary 5.2.5 (Unconditional convergence of Bessel sequences)LET fk k∈N ⊂ Hn be a Bessel sequence.

THEN∑

k∈N ck fk converges unconditionally for all ck k∈N ∈ l 2n(N).

Lemma 5.2.6 (Bessel sequence on a dense set)LET fk k∈N ⊂ Hn and suppose there exists a dense subset V ⊂ Hn and a constant B > 0 suchthat ∑

k∈N|⟨⟨ f , fk

⟩⟩|2 ≤ B‖ f ‖2, ∀ f ∈V.

THEN fk k∈N is a Bessel sequence in Hn with bound B .

Proof. The topology of the Clifford-Hilbert module Hn is defined by the Euclidean norm andthus identical to the topology on the complex Hilbert space

(Hn , (·, ·)). Thus, assume

∃g ∈ Hn :∑

k∈N

∣∣⟨⟨g , fk⟩⟩∣∣2 > B‖g‖2.

Then there exists a finite set

F ⊂N :∑

k∈F

∣∣⟨⟨g , fk⟩⟩∣∣2 > B‖g‖2.


Since ∣∣⟨⟨·, fk⟩⟩∣∣2 =

⟨⟨⟨·, fk⟩⟩⟨⟨

fk , ·⟩⟩⟩0

,

we know that the operator g 7→ ∑k∈F

∣∣⟨⟨g , fk⟩⟩∣∣2 is continuous. Since V is dense in Hn this

implies∃h ∈V :

∑k∈F

∣∣⟨⟨h, fk⟩⟩∣∣2 > B‖h‖2.

But this is a contradiction.

5.2.2 Clifford frames

Definition 5.2.7 (Clifford Frame)LET fk k∈N ⊂ Hn be a sequence such that there exist constants 0 < A ≤ B < ∞, satisfying theframe inequality

A‖ f ‖2 ≤ ∑k∈N

∣∣⟨⟨ f , fk⟩⟩∣∣2 ≤ B‖ f ‖2, ∀ f ∈ Hn . (5.2)

THEN fk k∈N is called a Clifford frame for Hn . A is called a lower frame bound, B is calledan upper frame bound.

• A Clifford frame is called tight, iff choosing A = B is possible.

• Let T be the synthesis operator of the frame fk . Then the Clifford frame operator isdefined by

S : Hn 7→ Hn , f 7→ S f := T T ∗ f = ∑k∈N

⟨⟨f , fk

⟩⟩fk .

• A sequence gk ∈N ⊂ Hn is called a dual frame of the Clifford frame fk k∈N iff gk k∈Nis a Clifford frame and the frame decompostion

f = ∑k∈N

⟨⟨f , fk

⟩⟩gk = ∑

k∈N

⟨⟨f , gk

⟩⟩fk

holds for all f ∈ Hn .

Remark 5.2.8 (Frame operator and frame inequality)For frames on Hilbert spaces the equality∑

k|⟨ f , fk⟩|2 = ⟨S f , f ⟩, ∀ f ∈ H (5.3)

is very useful. Note that for Clifford frames in general⟨⟨S f , f

⟩⟩=∑k

⟨⟨f , fk

⟩⟩⟨⟨fk , f

⟩⟩ 6=∑k

∣∣⟨⟨ f , fk⟩⟩∣∣2.

For Clifford frames equation (5.3) is replaced by∑k

∣∣⟨⟨ f , fk⟩⟩∣∣2 = (S f , f ), ∀ f ∈ Hn .

That is for Clifford frames the interplay between inner product ( · , · ) and Clifford algebravalued product

⟨⟨ · , · ⟩⟩on Hn is essential.


EXAMPLE 5.2.1 (A Clifford Frame forH):i is a tight Clifford frame for the quaternions. Let 0 6= f = fr + i fi + j f j + k fk ∈ H. Then|⟨⟨ f , i

⟩⟩| = |i fr + fi +k f j − i fk | = | f |. A corresponding dual frame is −i . Note that i is not aframe for the real Hilbert space

(H, (·, ·)).

EXAMPLE 5.2.2:LET fk k∈N be a frame for a Hilbert space H .

THENe0 fk

k∈N is a Clifford frame for Hn .

Proof. To verify thate0 fk

k∈N is a Clifford frame, let g =∑

α∈Oneαgα ∈ Hn . Then

∑k∈N

∥∥∥⟨⟨g ,e0 fk

⟩⟩∥∥∥2 = ∑k∈N

∥∥∥⟨⟨ ∑α∈On

eαgα,e0 fk⟩⟩∥∥∥2

=∑k

∥∥∥ ∑α∈On

eα⟨gα, fk⟩∥∥∥2 = ∑

α∈On

∑k

∣∣∣⟨gα, fk⟩∣∣∣2

≤ B∑

α∈On

‖gα‖2H

= B‖g‖2Hn

.

A lower frame bound can be derived in a similar way.

Lemma 5.2.9 (Properties of the frame operator and the canonical dual frame)LET fk k∈N ⊂ Hn be a Clifford frame with frame operator S and frame bounds A,B .

THEN

(i) The frame operator S is bounded, self-adjoint, invertible and positive. ( An operator iscalled positive, iff ∃0 < A : A‖ f ‖2 ≤ (S f , f ).)

(ii) S−1 fk k∈N is a Clifford frame, called the canonical dual frame, with frame boundsB−1, A−1. The frame operator for

S−1 fk

k∈N is S−1.

Proof. ad(i) S is bounded as it is the composition of bounded operators. By Theorem 5.2.3

‖S‖ = ‖T T ∗‖ = ‖T ‖‖T ∗‖ = ‖T ‖2 ≤ B.

Since S∗ = (T T ∗)∗ = T T ∗ = S the frame operator is self-adjoint.

Now

(S f , f ) =( ∑

k∈N

⟨⟨f , fk

⟩⟩fk , f

)=

⟨⟨⟨ ∑k∈N

⟨⟨f , fk

⟩⟩fk , f

⟩⟩⟩0

3.2.4(i )= ∑k∈N

⟨⟨⟨f , fk

⟩⟩⟨⟨fk , f

⟩⟩⟩0

3.2.4(i v)= ∑k∈N

∣∣⟨⟨ f , fk⟩⟩∣∣2

From (5.2) we know that (S f , f ) is positive and bounded for all f ∈ Hn . We write

A Id ≤ S ≤ B Id. (5.4)


Since S is positive, we know that 0 ≤ Id−B−1S ≤ B−AB Id < Id. That is,

‖ Id−B−1S‖ = sup‖ f ‖≤1

∣∣((Id−B−1S) f , f)∣∣≤ B − A

B< 1.

Hence S is invertible by its Neumann series. (See [29] chapter I, example 4.5.)

Furthermore, S−1 is computed using a series of positivCn-left-linear operators, and con-sequently S−1 is positive andCn-left-linear.

ad(ii) ∑k∈N

∣∣⟨⟨ f ,S−1 fk⟩⟩∣∣2 = ∑

k∈N

∣∣⟨⟨S−1 f , fk⟩⟩∣∣2 ≤ B‖S−1 f ‖2

≤ B‖S−1‖2‖ f ‖2, ∀ f ∈ Hn .

ThusS−1 fk

k∈N is a Bessel sequence. That is, its frame operator is well defined. Now,

since S isCn-left-linear, so is S−1 and hence∑k∈N

⟨⟨f ,S−1 fk

⟩⟩S−1 fk = S−1

∑k∈N

⟨⟨S−1 f , fk

⟩⟩fk = S−1SS−1 f = S−1 f . (5.5)

That is, the frame Operator ofS−1 fk

k∈N is S−1.

To compute the frame bounds forS−1 fk

k∈N, we multiply (5.4) by S−1 to obtain

AS−1 ≤ Id ≤ BS−1.

That is,B−1‖ f ‖2 ≤ (S−1 f , f ) ≤ A−1‖ f ‖2, ∀ f ∈ Hn .

Since S−1 is the frame operator ofS−1 fk

k∈N, it follows by (5.5) that

B−1‖ f ‖2 ≤ ∑k∈N

∣∣⟨⟨ f ,S−1 fk⟩⟩∣∣2 ≤ A−1‖ f ‖2, ∀ f ∈ Hn .

The next theorem shows that the canonical dual frame defined in Lemma 5.2.9 is indeed a dualframe according to the definition in Definition 5.2.7.

Theorem 5.2.10 (Existence of a dual frame)LET fk k∈N ⊂ H be a Clifford frame with frame operator S. Then

f = ∑k∈N

⟨⟨f ,S−1 fk

⟩⟩fk , ∀ f ∈ Hn ,

andf = ∑

k∈N

⟨⟨f , fk

⟩⟩S−1 fk , ∀ f ∈ Hn .

Both series converge unconditionally for all f ∈ Hn .

Proof. Let f ∈ Hn . On the one hand

f = SS−1 f = ∑k∈N

⟨⟨S−1 f , fk

⟩⟩fk = ∑

k∈N

⟨⟨f ,S−1 fk

⟩⟩fk ,


and on the other hand

f = S−1S f = S−1∑

k∈N

⟨⟨f , fk

⟩⟩fk = ∑

k∈N

⟨⟨f , fk

⟩⟩S−1 fk .

The unconditional convergence follows since fk k∈N and S−1 fk k∈N are Bessel sequencesand T ∗ f ∈ l 2

n(N).

Remark 5.2.11 (Existence of a frame decomposition)The central property of Clifford frames is that they allow a basis-like decomposition – the framedecomposition – of a function in a Clifford-Hilbert module into a series in l 2

n(N) by the anal-ysis operator and reconstruction from this series via the synthesis operator of a dual frame.Theorem 5.2.10 shows that at least one dual frame exists – the canonical dual frame.

Lemma 5.2.12 (Frames on dense sets)LET fk k∈N ⊂ H . If there exist 0 < A ≤ B <∞ such that for a dense subset V ⊆ Hn

A‖ f ‖2 ≤ ∑k∈N

∣∣⟨⟨ f , fk⟩⟩∣∣2 ≤ B‖ f ‖2, ∀ f ∈V.

THEN fk k∈N is a Clifford frame for Hn with frame bounds A,B .

Proof. In Lemma 5.2.6 we showed that the upper bound holds. Now we know by Lemma 5.2.9(ii)that for the dual frame S−1 fk it is true that

∑k∈N

∣∣⟨⟨ f ,S−1 fk⟩⟩∣∣2 ≤ 1

A‖ f ‖2 ∀ f ∈V.

Hence by Lemma 5.2.6 ∑k∈N

∣∣⟨⟨ f ,S−1 fk⟩⟩∣∣2 ≤ 1

A‖ f ‖2 ∀ f ∈Hn ,

and again by Lemma 5.2.9(ii) the lower bound follows.

Theorem 1.3.27 on page 13 shows that a surjective operator maps a frame to a frame of itsrange. The next theorem is the analog of Theorem 1.3.27 for Clifford frames. In Example 5.1.2we used Theorem 1.3.27 to prove that the Hilbert transform of a frame is once again a frame.The next theorem will play this role for the Riesz transform of a frame.

Theorem 5.2.13 (Clifford frames and operators)LET fk k∈N ⊂ Hn be a Clifford frame for Hn with bounds A,B . Let V ⊆ Hn . Furthermore, letU : Hn 7→V be a bounded surjective operator.

THEN U fk k∈N is a Clifford frame for V with bounds A‖U †‖−2,B‖U‖2. Here U † =U∗(UU∗)−1

denotes the pseudoinverse.

Proof. Let f ∈V . Then ∑k∈N

∣∣⟨⟨ f ,U fk⟩⟩∣∣2 ≤ B‖U∗ f ‖2 ≤ B‖U‖2‖ f ‖2,

whence U fk k∈N is a Bessel sequence.

Since U is surjective, there exists a g ∈ Hn such that f =Ug .

5.3. MONOGENIC WAVELETS 87

Let U † := U∗(UU∗)−1 be the pseudo-inverse of U . For details on the pseudo-inverse of anoperator on a Hilbert space see [10] A.7.

Now UU † =UU∗(UU∗)−1 = IdHn is self adjoint. Therefore

f =Ug = (UU †)∗Ug = (U †)∗U∗Ug .

And hence

‖ f ‖2 ≤ ‖(U †)∗‖2‖U∗Ug‖2 ≤ A−1‖(U †)∗‖2∑

k∈N

∣∣⟨⟨U∗Ug , fk⟩⟩∣∣2

= A−1‖U †‖2∑

k∈N

∣∣⟨⟨ f ,U fk⟩⟩∣∣2.

Now Lemma 5.2.12 yields the frame property on V .

5.3 Monogenic wavelets

In the last section we have made the first step towards extending the analytical wavelets in-troduced in Example 5.1.2 to higher dimensions. In this section we will introduce a suit-able wavelet transform and finish our agenda to define monogenic wavelet frames by Defi-nition 5.3.2.

Definition 5.3.1 (Hypercomplex wavelet transform)LET ψ ∈ L2(Rn)n and let D be a dilation matrix.

THEN

D j Tkψ j ,k

is called a wavelet system, where j ∈Z and k ∈Zn or j ∈R and k ∈Rn with mother waveletψ.

LET f ∈ L2(Rn)n .

THEN

Wψ( f ) :=⟨⟨

f ,D j Tkψ⟩⟩

j ,k

is called the hypercomplex wavelet transform of f .

Definition 5.3.2 (Monogenic wavelet)LET ψ ∈ L2(Rn) be a mother wavelet for L2(Rn) with respect to a dilation matrix D.

THEN the monogenic wavelet system D j Tkψm j ,k corresponding to ψ is generated by themonogenic mother wavelet

ψm :=ψ+Rψ=ψ+∑eαRαψ,

where R is the hypercomplex Riesz transformation defined in Definition 4.1.1 on page 66. Anelement D j Tkψ of the monogenic wavelet system is called a monogenic wavelet. The hyper-complex wavelet transform Wψm is called the monogenic wavelet transform.


5.3.1 Monogenic wavelet frames

Theorem 5.3.3 (Riesz transforms of frames)LET fk k∈N be a frame for L2(Rn ,R) with frame bounds A and B .

THEN the following hold:

(i) fk k∈N is a Clifford frame for L2(Rn ,C)n∼= L2(Rn ,Cn) with the same frame bounds A

and B .

(ii) The Riesz transformed frame R fk k∈N is a Clifford frame for L2(Rn ,C)n with the sameframe bounds A and B .

Proof. Let fk k∈N be a frame for L2(Rn ,R). Then e0 fk k∈N is a frame for L2(Rn ,Cn). Toverify this, let g =∑

α∈Oneαgα ∈ L2(Rn ,Cn). Thus

∑k∈N

∥∥∥⟨⟨g ,e0 fk

⟩⟩∥∥∥2 = ∑k∈N

∥∥∥⟨⟨ ∑α∈On

eαgα,e0 fk⟩⟩∥∥∥2

=∑k

∥∥∥ ∑α∈On

eα⟨gα, fk⟩∥∥∥2 = ∑

α∈On

∑k

∣∣∣⟨gα, fk⟩∣∣∣2

≤ B∑

α∈On

‖gα‖2L2(Rn ,C)

= B‖g‖2L2(Rn ,C)n

.

A lower frame bound can be derived in a similar way. This proves (i).

The Riesz transform is its own inverse, hence invertible and therefore surjective in L2(Rn ,Cn).(See Theorem 4.1.4 on page 66.) Now statement (ii) follows by Theorem 5.2.13.

Summing up the Riesz transform of a frame for L2(Rn ,R) yields a Clifford frame for L2(Rn ,Cn).

Theorem 5.3.4 (Monogenic wavelet frame)LET ψ ∈ L2(Rn) generate a wavelet frame for L2(Rn) with respect to a dilation matrix D withframe bounds 0 < m ≤ M <∞.

THEN the monogenic wavelet generates a wavelet frame for T H 2−(Rn) with frame bounds 2mand 2M .

LET φ ∈ L2(Rn) generate a dual frame for the frame with respect to the same dilation andtranslation.

THEN the dual frame of the monogenic wavelet frame with mother wavelet ψm is the framewith mother wavelet 1

2φm .

LET the dilation matrix constitute a rotated dilation, i.e. D = Dd := Ddρ, where d > 1 andρ ∈ SO(n).

THEN the wavelet transform of a function in L2(Rn) satisfies

Wψm f (t , j ) =Wψ f (t , j )+∑l ,k

el (ρ j )k,l WψRk f (t , j ).

Thus the monogenic signal of a function f ∈ L2(Rn) may be computed from the coefficients ofthe monogenic wavelet.

5.3. MONOGENIC WAVELETS 89

Proof. Sinceψ generates a frame for L2(Rn) with real frame coefficients, using Clifford algebravalued frame coefficients it generates a frame for L2

n(Rn). T H 2−(Rn) is the image of eoL2(Rn)under the monogenic signal and by Theorem 5.3.3 ψm is a mother wavelet for a wavelet frameof T H 2−(Rn).

Let f ∈ L2n(Rn), t ∈Zn and j ∈Z.

Wψm f (t , j ) = ⟨⟨f ,D j

d Ttψm⟩⟩

= ⟨⟨f ,D j

d Tt (ψ+Rψ)⟩⟩

= ⟨ f ,D jd Ttψ⟩+

n∑l=1

el i⟨⟨

f ,Dd j ρ j Tt Rlψ⟩⟩


n∑l ,k=1

(ρ j )k,l el i ⟨ f ,Rk Dd j ρ j Ttψ⟩


n∑l ,k=1

(ρ j )k,l

∫Rn

el f (ξ)iξk

|ξ|F(Dd j ρ j Ttψ

)(ξ)dξ


n∑l ,k=1

(ρ j )k,l

∫Rn

el i f (ξ)F(D

jd Ttψ

)(ξ)

ξk

|ξ|dξ


n∑l ,k=1

(ρ j )k,l el ⟨Rk f ,D jd Ttψ⟩.

Here we used Theorem 2.1.2 (i, ii, iii, vii) on page 20.

We now show that ∀ f , g ∈ L2(Rn) the following identity holds( ∑j∈Z,t∈Zn

⟨⟨e0 f ,D j

d Ttψm⟩⟩D

jd Ttφm ,e0g

)= 2⟨ f , g ⟩,

using what we have just proven.

( ∑j∈Z,t∈Zn

⟨⟨e0 f ,D j

d Ttψm⟩⟩D

jd Ttφm ,e0g

)=∑

j ,t

⟨⟨⟨e0 f ,D j

d Ttψm⟩⟩⟨⟨

Djd Ttφm ,e0g

⟩⟩⟩0

=∑j ,t

⟨ f ,D jd Ttψ⟩⟨D j

d Ttφ, g ⟩+⟨ n∑

k,l=1(ρ j )k,l el ⟨Rk f ,D j

d Ttψ⟩n∑

r,s=1(ρ j )r,s es⟨D j

d Ttφ,Rr g ⟩⟩

0

=⟨∑

j ,t⟨ f ,D j

d Ttψ⟩D jd Ttφ, g

⟩+∑

j ,t

n∑k,r,l=1

(ρ j )k,l (ρ j )r,l︸︷︷︸=(ρρT )k,r =Idk,r =δk,r

⟨Rk f ,D jd Ttψ⟩⟨D j

d Ttφ,Rr g ⟩

= ⟨ f , g ⟩+n∑

k=1

∑j ,t

⟨Rk f ,D jd Ttψ⟩⟨D j

d Ttφ,Rk g ⟩

= ⟨ f , g ⟩+n∑

k=1

⟨∑j ,t

⟨Rk f ,D jd Ttψ⟩D j

d Ttφ,Rk g⟩

= ⟨ f , g ⟩+n∑

k=1⟨Rk f ,Rk g ⟩

= 2⟨ f , g ⟩, ∀ f , g ∈ L2(Rn).


5.4 Decay of monogenic wavelets

The joint localization of wavelets in space and frequency is one of their key properties. As afirst step we now look at the decay rate of Riesz transformed functions. The Riesz transformleaves the decay in the Fourier domain invariant. In the following we derive novel conditionsunder which a partial Riesz transform of a function inherits the decay rate.

We take a closer look at the joint space-frequency localization of the monogenic wavelets whenwe consider the effect of the Riesz transform on uncertainty relations in Theorem 6.3.12.

Lemma 5.4.1 (Smoothness and decay)LET k ∈N0 and f ∈S ′(Rn).

THEN

∂β f ∈ L2(Rn), ∀|β| ≤ k iff∫Rn

| f (ξ)|2(1+|ξ|2)k dξ<∞.

Here the derivative operator should be considered in the distributional sense.

Proof. The proof can be found in [20], Chapter 1.2.

Proposition 5.4.2LET f ∈C k (Rn) and furthermore let f (α)(0) = 0, ∀|α| ≤ k.

THEN Rα f ∈C k (Rn).

Proof. Remember from Theorem 2.2.8 on page 30 that the partial derivatives of the Fouriermultiplier of the Riesz transform are of the form

∂βξα

|ξ| =hα,β(ξ)

|ξ||β| , ∀ξ ∈Rn \ 0, (5.6)

where hα,β :Rn →R is a 0-homogenous function.

To show that Rα f ∈ C k (Rn) we have to show that all partial derivatives ∂βRα f , |β| ≤ k, arecontinuous.

Since ∂βRα ∈C∞(Rn \ Bε(0)), ∀ε> 0 it is sufficient to show that

limx→0

∂βRα f (ξ) = 0, ∀|β| ≤ k.

For ξ ∈Rn , the product rule yields

∂βRα f (ξ) =n∑

γ≤β

(n

γ1

)· · ·

(n

γn

)∂γξα

|ξ|∂β−γ f (ξ),

where γ= (γ1, · · · ,γn) ∈Nn0 .

Combined with Equation 5.6 this yields a term of the form

∂βRα f (ξ) = ∑γ≤β

(n

γ1

)· · ·

(n

γn

)hα,γ(ξ)

|ξ||γ| ∂β−γ f (ξ), ∀ξ ∈Rn \ 0.

By Taylors Theorem A.1.3 on page 166 and since ∂ν f (0) = 0, ∀|ν| ≤ k, we know that

0 = limξ→0

∂β−γ f (ξ)−T|γ|∂β−γ f (ξ;0)

|ξ||γ| = limξ→0

∂β−γ f (ξ)

|ξ||γ| ,

5.5. EXPLICIT CONSTRUCTION OF WAVELET FRAMES IN L2(RN ) 91

where T|γ|∂β−γ f (ξ; a) =∑|γ|l=0

1l !

∑α∈Nn

0|α|=l

∂α+β−γ f (0)ξα.

Theorem 5.4.3 (Vanishing moments and decay of the Riesz transform)LET k ∈ N0. Furthermore let f ∈ L2(Rn) be such that ∂β f ∈ L2(Rn), ∀|β| ≤ k and f ∈ C k .Finally, let ∂β f (0) = 0, ∀|β| ≤ k.

THEN Rα f ∈W k,2(Rn). I.e. ∫Rn

|Rα f (ξ)|2(1+|ξ|2)k dξ<∞.

Proof. We want to use Lemma 5.4.1. Hence we need to proof that

∂βRα f ∈ L2(Rn), ∀|β| ≤ k.

In Proposition 5.4.2 we proved the existence of the derivatives. Square-integrability followseasily if we consider the integrability on B1(0) and on Rn \ B1(0) separately. We have alreadyshown, that ∂βRα f is continuous on the compact set B1(0), hence boundedness and square-integrability follows. As we have seen in Remark 2.2.7 on page 29

∂βRα f (ξ) = ∑γ≤β

(n

γ1

)· · ·

(n

γn

)∂γ f (ξ)

pα,γ(ξ)

q2|γ|+1(|ξ|) , ∀ξ ∈Rn .

But ∂γ f ∈ L2(Rn) andpα,γ(ξ)

q2|γ|+1(|ξ|) is dominated by a finite constant on (Rn)\B1(0). Hence square-

integrability follows.

Remark 5.4.4An important condition of Theorem 5.4.3 is the existence of the derivatives of f of order less orequal to k. The existence of the derivatives, at least in a neighborhood of 0, is indeed necessaryfor our proof. It would be sufficient to have weak derivatives of degree k on the rest of Rn .

5.5 Explicit construction of wavelet frames in L2(Rn)

On our way to implement the monogenic signal via wavelet frames we have found in Theo-rem 5.3.4 that we can derive a monogenic wavelet frame from a real wavelet frame of L2(Rn).Hence our next step should be to construct suitable wavelet frames for L2(Rn).

The present section presents a suitable way to construct wavelet frames for L2(Rn) with re-spect to a given isotropic or rotated dilation.

5.5.1 Wavelets on irregular grids with arbitrary dilation matrices, and frameatoms for L2(Rn)

This section introduces a construction principle for frames of L2(Rn) based on [2]. We finishthe section with our own Corollary 5.5.6 that shows how to acquire tight wavelet frames.

Proposition 5.5.1LET U ,V be non-empty, open subsets of Rn , and let T : U 7→ V be a diffeomorphism, i.e.,


T ∈C 1(Rn) and T −1 ∈C 1(Rn), such that

0 <α := infy∈U

|det(T ′(y)|−1 ≤β := supy∈U

|det(T ′(y))|−1 <∞,

where T ′(y) :=(∂Ti∂y j

(y))

i , j. Furthermore, let g j j∈Z be a frame for

KV := f ∈ L2(Rn) : supp( f ) ⊆V

with lower frame bound m and and upper frame bound M such that 0 < m < M <∞.

THEN g j T j∈Z is a frame for KU := f ∈ L2(Rn) : supp( f ) ⊆U

with lower frame bound αm

and upper frame bound βM .

Proof. Let f ∈ KU . We will write S := T −1, with this notation

⟨ f , g j T ⟩ =∫

Uf (t )g j (T (t ))d t =

∫V

f (S(s))|det(S′(s))|g j (s)d s

= ⟨|det(S′(.))| f S, g j ⟩Now, |det(S′(.))| f S ∈ KV , since 0 < |det(S′(s))| <∞, ∀s ∈V . It follows that

m‖|det(S′(.))| f S‖22 ≤

∑j∈Z

|⟨|det(S′(.))| f S, g j |2 ≤ M‖|det(S′(.))| f S‖22.

Furthermore,

‖|det(S′(.))| f S‖22 =

∫V|det(S′(t ))|2| f (S(t ))|2d t

=∫

U|det(S′ T (s))|2|det(T ′(s))|| f (s)|2d s.

Now since |det(S′ T (.))| = (|det(T ′(.))|)−1 and α‖ f ‖2 ≤ ∫U |det(T ′(.))|−1| f (t )|2 ≤ β‖ f ‖2d t it

follows thatmα‖ f ‖2 ≤ ∑

j∈Z|⟨ f , g j T ⟩|2 ≤βM‖ f ‖2.

Corollary 5.5.2LET Q ⊆Rn be an open subset, y ∈Rn and let A ∈GLn(Rn) be invertible.

THEN g j j∈Z is a frame for KQ , with frame bounds m, M , iff

(i) Ty g j j∈Z is a frame for KT−y Q with frame bounds m, M .

(ii) Ag j j∈Z is a frame for K A−1Q with frame bounds m, M .

Definition 5.5.3 (Riesz partition of unity)LET h j j∈Z ⊂ L2(Rn) be a set of functions in L2(Rn) such that there exist positive finite numbers0 < p ≤ P <∞ such that

p ≤ ∑j∈Z

|h j (x)|2 ≤ P, a.e. x ∈Rn . (5.7)

THEN h j j∈Z ⊂ L2(Rn) is a Riesz partition of unity.

LET p = P = 1.

THEN H = h j j∈Z is called a regular partition of unity.


The next theorem reduces the problem to construct a frame for L2(Rn) to the problem of find-ing a Riesz partition of unity.

Theorem 5.5.4 (Frames and Riesz partitions of unity)LET H = h j j∈Z ⊂ L2(Rn) be a Riesz partition of unity with bounds p,P and denote

W j := h j f : f ∈ L2(Rn).

Furthermore let g j ,k k be a frame for W j with frame bounds m j , M j and let

0 < m = infj

m j ≤ supj

M j = M <∞.

THEN h j g j ,k : j ,k ∈Z is a frame for L2(Rn) with frame bounds pm and P M .

Proof. Let f ∈ L2(Rn) and set f j := h j f ∈W j . Then

mp‖ f ‖2 = infj

m j

∫Rn

p| f (t )|2d t ≤ infj

m j

∫Rn

∑j|h j (t )|2| f (t )|2d t

= infj

m j

∫Rn

∑j|h j (t ) f (t )|2d t ≤∑

jm j ‖ f j (t )‖2

≤∑j

∑k|⟨ f ,h j g j ,k⟩| ≤

∑j

M j ‖ f j (t )‖2

≤ supj

M j

∫Rn

∑j|h j (t )|2| f (t )|2d t ≤ sup

jM j

∫Rn

P | f (t )|2d t

=MP‖ f ‖2.

Remark 5.5.5At first view Theorem 5.5.4 does not seem to make the problem of finding a wavelet framefor L2(Rn) any easier. If however the Riesz partition of unity is chosen as dilates of a singlecompactly supported function h ∈ L2(Rn), then the g j ,k k∈Z may be chosen as dilates of a set

of modulations and àh j g j ,k j ,k∈Z is a wavelet frame for L2(Rn).

Since we wish to construct wavelet frames we have to make sure that the dual frame is a waveletframe, too. The easiest way to do this is to construct a tight wavelet frame. The Riesz partitionof unity H and the frame g j ,k k∈Z for the corresponding W j can be chosen to yield a tightwavelet frame as follows.

Corollary 5.5.6 (Tight wavelet frames from regular partitions of unity)LET h ∈ L2(Rn) and let H = h j := h(A j ·) : j ∈ Z be a regular partition of unity, where A is adilation. Let

W j := h j f : f ∈ L2(Rn).

Furthermore, let w > 0 be such that supp(h) ⊆ [− 12 w, 1

2 w]n . Finally, let

g j ,k (x) := A j g0,k (x) := 1

wn A j e−2π

w i ⟨x,k⟩.

THEN g j ,k k is a tight frame for W j with frame bound 1. Furthermore, h j g j ,k j ,k∈Z is a tightframe for L2(Rn) with frame bound 1 and its Fourier transform F (h j g j ,k ) j ,k∈Z is a wavelettight frame for L2(Rn) with frame bound 1.


Proof. e−2πi ⟨x,k⟩k∈Z is an orthonormal basis for L2(− 12 , 1

2 ) (see for example [41]) and hence atight frame with bound 1 for every subset of L2(− 1

2 , 12 ) . Thus by Corollary 5.5.2 g0,k k∈Z is a

tight frame with bound 1 for W0. Again by Corollary 5.5.2 g j ,k k∈Z is a tightframe with bound 1for W j . Now Theorem 5.5.4 proves that h j g j ,k : j ,k ∈Z is a tight frame for L2(Rn) with framebound 1.

By Theorem 1.3.22 on page 12 we know that

F (h j g j ,k ) =F(h(A j ·) 1

wn A j e−2πi ⟨·,k⟩/w )= 1

wn F (A j M−k/w h) = 1

wn (AT )− j T−k/w h,

whence F (h j g j ,k ) : j ,k ∈Z is a wavelet tight frame for L2(Rn) with frame bound 1.

5.5.2 Construction of regular partitions of unity

LetT := [− 12 , 1

2 ]n and wT := [− 12 w, 1

2 w]n . Corollary 5.5.6 states that, in order to construct thewavelet frames with respect to a dilation matrix A, it is sufficient to construct a regular partitionof unity of the form H = h(A j ·) j∈Z such that supp(h) ⊆ wT, for some w > 0.

The following new theorem reduces the problem of finding a regular partition of unity to thatof finding a certain function which then allows to easily control differentiability.

Theorem 5.5.7 (Construction of regular partitions of unity)For a closed set A denote by A := ⋃

O ⊂ A : O is open

the interior and by ∂A := A \ A theboundary of A.

LET S ⊆T be a closed nonempty set and let A be a dilation matrix onRn such that⋃j∈Z

A j S =Rn \ 0 and S ∩ AS =;.

We set S := ∂S ∩∂A−1S and suppose that ∂S = S⊕ AS.

Let f ∈C l (Rn), l ∈N0 be such that 0 ≤ f (x) ≤ 1/4,∀x ∈ S, and furthermore let on the boundary

f (x) = 0, f (Ax) = 1/4 ,∂α f (x) = ∂α f (Ax) = 0, ∀α ∈Nn0 : 1 ≤ |α| ≤ l , ∀x ∈ S.

THEN

h(x) =

cos(2π f (Ax)), ∀x ∈ A−1S;

sin(2π f (x)), ∀x ∈ S;

0, otherwise,

defines a regular partition of unity with respect to the dilation A by

H = h(A j ·) j∈Z.

Furthermore, h j = h(A j · ) ∈C l (Rn), ∀ j ∈Z.

Proof. We first show that H is a regular partition of unity. Let x ∈Rn . Then ∃k ∈Z : x ∈ A−k S.Thus, ∑

j∈Z|h j (x)|2 = |cos(2π f (Ak−1(Ax)))|2 +|sin(2π f (Ak x))|2 = 1.


Note that ∂S = A−1S+S and ∂A−1S = A−2S+ A−1S. Now, ∀x ∈ S : h(A−2x) = cos(π/2) = 0,and h(x) = sin(0) = 0, and h(A−1x) = cos(0) = sin(π/2) = 1. Thus h is well defined and continu-ous.

To show that h ∈C l (Rn), we need to show that ∂αh is a continuous function for allα ∈Nn0 such

that |α| ≤ l , x ∈ A−1S +S, especially in ∂S and ∂A−1S.

∂αh is of the form

∂αh(x) =

l∑k=1

∑β≤α|β|=k

cβ∂α−β

( n∏m=1

(1−δβm ,0)∂βm (2π f (Ax))

∂xm

)cos(k)(2π f (Ax)), ∀x ∈ A−1S,

l∑k=1

∑β≤α|β|=k

cβ∂α−β

( n∏m=1

(1−δβm ,0)∂βm (2π f (x))

∂xm

)sin(k)(2π f (x)), ∀x ∈ S,

0, otherwise,

where cβ is a positive integer,α−β= (α1−β1, . . . ,αn −βn) and β≤α iff βm ≤αm , ∀m = 1, . . . ,n.The term (1−δβm ,0) guarantees that only terms with βm > 0 are considered.

The sum contains only terms of the form t1(x) = ∂β(2π f (Ax)) times t2(x) = cos(k)(2π f (Ax)) forx ∈ A−1S, respectively t1(x) = ∂β(2π f (x)) times t2(x) = sin(k)(2π f (x)) for x ∈ S, where 1 ≤ |β| ≤ land 1 ≤ k ≤ l . By assumption, t1(x) = t1(A−1x) = t1(A−2x) = 0, ∀x ∈ S and the factor t2 isbounded. Hence ∂αh(x) = ∂αh(Ax) = ∂αh(A2x) = 0, ∀x ∈ S.

5.5.3 An example of a tight wavelet frame

EXAMPLE 5.5.1 (A radial tight wavelet frame):Let us consider the problem of constructing a radial wavelet frame. Then the canonical choicefor S is

S =

x ∈Rn :1

4≤ |x| ≤ 1

2

.

The problem to find a regular partition of unity of C l (Rn)-functions is now reduced to find-ing a function in C l ( 1

4 , 12 ) such that f (| · |) satisfies the requirements of Theorem 5.5.7. We

now look at examples of such functions f that give a regular partition of unity of R with re-spect to dyadic dilations. If l ∈N0, we can easily find polynomials pl satisfying the conditionspl ( 1

4 ) = 1/4, pl ( 12 ) = 0 and p(k)(m) = 0, ∀m = 1

4 , 12 , 0 < k ≤ l . Since a polynomial of degree one,

namely p1(x) = 1/2− x is required to satisfy p1( 14 ) = 1/4 and p1( 1

2 ) = 0, p1 has a non-vanshing

derivative, whence we require the polynomials to have degree 2l −1 to obtain h ∈C l (R). Fur-ther examples for polynomials pl resulting in functions h ∈C l (R) are

p0(x) =1/2−x (5.8)

p1(x) =−1+12x −36x2 +32x3

p2(x) =8−120x +720x2 −2080x3 +2880x4 −1536x5

p3(x) =−52+1120x −10080x2 +49280x3 −141120x4 +236544x5 −215040x6 +81920x7

p4(x) =368−10080x +120960x2 −833280x3 +3628800x4 −10354176x5 +19353600x6

−22855680x7 +15482880x8 −4587520x9

p5(x) =−2656+88704x −1330560x2 +11827200x3 −69189120x4 +279595008x5 −796207104x6

+1597685760x7 −2214051840x8 +2018508800x9 −1089994752x10 +264241152x11


0.30 0.35 0.40 0.45 0.50

0.05

0.10

0.15

0.20

0.25

Figure 5.2: The functions p0 (light brown), p1 (green), p2 (light blue), p3 (dark blue), p4 (violet),p5 (red) and f6 (black).

Now we give an example for a function λ ∈C∞(1,2) that yields a h ∈C∞(R).

Let λ :R→R, x 7→

0, ∀x ≤ 0

e−x−2, ∀x ∈R+.

. Then λ ∈ C∞(R) and 0 ≤ λ(x) ≤ 1 ∀x ∈R+. If we let

ε> 0, then

fε(x) = λ(ε( 12 −x))

4(λ(ε(x − 1

4 ))+λ(ε( 12 −x))

)is a suitable function.

5.5.4 An alternative construction for Riesz partitions of unity

In subsection 5.5.2 we provide a method for generating a regular partition of unity. Anotherway to generate a regular partition of unity is to transform an existing regular partition ofunity on the real line with respect to translation H = T j h j∈Z to a regular partition of unitywith respect to (dyadic) dilation. Every orthonormal system of translates of a single func-tion φ in L2(Rn) with compact support yields a suitable Riesz partition of unity – indeed∑

k |φ(ξ+k)|2 = 1 holds almost everywhere onR. (See [55] proposition 2.1.11 but note the dif-ferent definition of the Fourier transform.) Of course for every multiresolution analysis thereexists a scaling function φ ∈ L2(Rn) that yields an orthonormal system of translates. Thus,regular Riesz partitions of unity with respect to translation are easily available.


-0.4 -0.2 0.2 0.4

0.2

0.4

0.6

0.8

1.0

Figure 5.3: The function h corresponding to p0 (light brown), p1 (green), p2 (light blue), p3

(dark blue), p4 (violet), p5 (red) and f6 (black).

Theorem 5.5.8LET h ∈ L2(R) be such that supph = [1, l ] and such that H := h j j∈Z := T j h j∈Z, is a regularpartition of unity forR.

THEN h′ := ∑li=0 h(2−i x + i )χ[2i ,2i+1](x) defines a regular partition of unity of R+ with respect

to the dyadic dilation by H ′ := h′(2 j ·) j∈Z. A regular partition of unity of Rn \ 0 is given by

H ′ := h′(2 j | · |) j∈Z.

Proof. Let x ∈R+. Then

∑j∈Z

(h′(2 j x)

)2 = ∑j∈Z

l∑i=0

(h(2 j−i x + i )χ[2i ,2i+1](2 j x)

)2

m= j−i= ∑m∈Z

l∑i=0

(h(2m x + i )χ[2−m ,2−m+1](x)

)2

∗=l∑

i=0

(h(y + i )

)2

= 1,

where ∗ holds with y = 2m x ∈ [1,2], with m ∈Z : χ[2−m ,2−m+1](x) = 1.

This theorem yields a regular partition of unity forRn \ 0 by h′(|x|).

The weakness of this method is that it does not conserve continuity or differentibility of thefunction φ. However, this is easily mended.


(a) Real wavelet (b) Riesz transformed wavelet

(c) Amplitude (d) Phase and phase direction

Figure 5.4: Wavelet for L2(R2) in the case m = 2. (a) Real part of the monogenic wavelet Ψm ,i.e., the real waveletΨ. (b) Riesz transforms R1Ψ, R2Ψ of the waveletΨ. (c) Amplitude ‖Ψm‖.(d) Phase φ and phase direction (arrows) of the hypercomplex wavelet Ψm . The isotropy isclearly visible. The direction of the phase points from high (π ↔ white) to low (0 ↔ black)phase values.

Corollary 5.5.9LET h ∈ C l (R) be in compliance with the conditions of Theorem 5.5.8. Furthermore, letτ : [1,2] → [1,2] such that τ ∈C (l )(R), τ(1) = 1, τ(2) = 2 and τ(k)(2) = (21/2D1/2τ)(k)(2) ∀1 ≤ k ≤ l .

THEN h′ := ∑li=0 h(2−iτ(x)+ i )χ[2i ,2i+1](x) is a regular partition of unity of R+ with respect to

the dyadic dilation by H ′ := h′(2 j ·) j∈Z. Furthermore h′ ∈C l (R+).

For an arbitrary dilation parameter a a Riesz partition of unity G = g (a j ·) j∈Z is given by

choosing g = h((a −1)(x +1)−1) and defining g ′ =∑li=0 h(a−i x + (a −1)i )χ[ai ,ai+1](x).

5.5.5 Interpretation via refinable functions

Extension Principles

We now show that under certain conditions the wavelets that we constructed will satisfy theextension principle developed by Ron and Shen in [39]. It follows that these wavelets can beconstructed by an extended concept of MRA ( Multi Resolution Analysis). This yields theexistence of a scaling function φ.

In order to state the extension principle, we need some notation.

Definition 5.5.10


LET A be a dilation matrix.

THEN

1. a function φ ∈ L2(Rn) is called a refinable function, or scaling function, iff

∃H0 ∈ L2(T) : φ(Aξ) = H0(ξ)φ(ξ) a.e..

H0 is called a refinement mask.

2. the scaling function defines a space V0 := spank∈Zn Tkφ. It follows thatV j := A j V0 ⊂V j−1 and A jφ ∈V j , ∀ j ∈Z.

3. a set of functionsΨ= ψl Ll=1 ⊂ L2(Rn) such that

∀l = 1, . . . ,L∃Hl ∈ L∞(Tn) : ψl (Aξ) = Hl (ξ)φ(ξ) a.e.

is called a wavelet system corresponding to the scaling function φ. The functions Hl are

called wavelet masks. We denote H :=

H0

H1...

HL

.

4. the fundamental function of multiresolution is defined as

Θ(ξ) := ∑r∈N0

L∑l=1

|Hl (Ar ξ)|2r−1∏j=0

|H0(A jξ)|2, ∀ξ ∈Rn \Zn .

5. the κ-function is defined by

κ :Rn →Z, ξ 7→ inf

j ∈Z : A jξ ∈Zn.

The following is Theorem 6.5 of [39].

Theorem 5.5.11LET A be an integer valued dilation matrix. Let φ be a refinable function, Ψ a finite set ofwavelets and H the corresponding refinement-wavelet mask. Assume that

(i) φ satisfies the mild decay condition,∑ψ∈Ψ

∑k∈N

∥∥ ∑a∈Ak

|ψ( · +a)|2∥∥L∞([−1/2,1/2]n ) <∞,

where Ak := a ∈Zn : |a| > 2k

; ( This is satisfied if ∃ρ > n

2 : ψ(ξ) =O(ξ−ρ) as ξ→∞.)

(ii) φ(0) = limξ→0 φ(ξ) = 1;

(iii) H is essentially bounded.

THEN Ψ generates a tight frame with bound C iff the following conditions hold

1. For a.e. ξ ∈Rn , lim j→−∞Θ(A jξ) =C .


2. For a.e. ξ,ω ∈Rn , if κ(ξ−ω) = 1, then

Θ(Aξ)H0(ξ)H0(ω)+L∑

l=1Hl (ξ)Hl (ω) = 0

unless φ vanishes identically on either ξ+Zn or ω+Zn .

In particular, in the case Θ = 1 a.e., A∗ j Tkψl j∈Z; k∈Zn , l=1,...,L is a tight frame if the vectors Hand TνH are perpendicular a.e., for every ν ∈ (A−1Zn/Zn) \ 0.

We now show that wavelet frames based on Theorem 5.5.7 satisfy the assumptions of the abovetheorem.

Corollary 5.5.12LET A be an integer valued dilation matrix and let f and S be as in Theorem 5.5.7.Set S∪ :=∪ j∈NA− j S.

THEN

g (ξ) :=

1, ∀ξ ∈ A−1S∪;

sin(2π f (Aξ)), ∀ξ ∈ A−1S;

0, otherwise,

defines a refinable function φ by φ= g .

Indeed a refinement mask is given by the 1-periodic function defined onT by

H0(ξ) :=

1, ∀ξ ∈ A−2S∪;

sin(2π f (A2ξ)), ∀ξ ∈ A−2S;

0, otherwise.

The refinement mask satisfies g (Aξ) = H0(ξ)g (ξ).

The corresponding wavelet mask is given by

H1(ξ) :=

0, ∀ξ ∈ A−2S∪;

cos2π f (A2ξ), ∀ξ ∈ A−2S;

1, ∀ξ ∈ A−1S;

0, otherwise.

The wavelet mask satisfies h(Aξ) = H1(ξ)g (ξ).

Proof. The fundamental function of multiresolution is in this case

Θ(ξ) = ∑r∈N0

|H1(Ar ξ)|2r−1∏j=0

|H0(A jξ)|2.

Now since H0(A jξ) = 1, ∀ξ ∈ suppH0(A j−1ξ), ∀ j ∈Z the product reduces to

r−1∏j=0

|H0(A jξ)|2 = |H0(Ar−1ξ)|2.


ThusΘ is bounded, since

Θ(ξ) = ∑r∈N0

|H1(Ar ξ)|2|H0(Ar−1ξ)|2

= ∑r∈N0

0, ∀ξ ∈ A−r−1S∪;

|cos(2π f (Ar+2ξ))|2, ∀ξ ∈ A−r−1S;

|sin(2π f (Ar+1ξ))|2, ∀ξ ∈ A−r S;

0, otherwise.

=

1, ∀ξ ∈ S∪;

|sin(2π f (Aξ))|2, ∀ξ ∈ S;

0, otherwise.

As a consequence, condition 1. of Theorem 5.5.11 is met, since

limn→−∞Θ(Anξ) = 1, ∀ξ ∈Rn \ 0.

To check condition 2., note thatΘ(Aξ) = 1, ∀ξ ∈ supp(H0). Furthermore κ(ω−ξ) = 1 is equiva-lent to ξ−ω ∈ A−1Zn/Zn . Hence we need to check if

H0(ξ)H0(ξ+q)+H1(ξ)H1(ξ+q) = 0, ∀ξ,ξ+q ∈ supp(g ), ∀q ∈ A−1Z

n \Zn .

This condition is automatically fulfilled for the dyadic dilation A = D2. This is because thedyadic dylation maps supp H1 = D−1

2 S ⊆ [− 14 , 1

4 ] to supp H1(D2·) = D−22 S ⊆ [− 1

8 , 18 ] and

|q |∞ = 2n+12 holds for q ∈ D−1

2 Zn/Zn and for some n ∈Z\ 1. It follows that

supp H1(D−12 ·)∩ suppTq H1(D−1

2 ·)

is a Lebesque zero set. Consequently


2 ·) =;.

Isotropic multi resolution analysis

The method of construction for radial wavelet tight frames we derived in Theorem 5.5.7 canalso be interpreted as a isotropic multiresolution analysis (IMRA). An introduction to IMRAcan be found in [38]. In [38] a different method to generate radial wavelet frames is considered.However, in contrast to our method, this does not yield a wavelet frame with only one motherwavelet.

Definition 5.5.13 (IMRA)An isotropic multiresolution analysis ( IMRA) of L2(Rn) with respect to a rotated dilationDa whose matrix entries are integer valued is a sequence V j j∈Z of closed subspaces of L2(Rn)satisfying the following conditions:

(1) ∀ j ∈Z, V j ⊂V j+1,

(2) DjaV0 =V j ,


(3)⋃j∈Z

V j is dense in L2(Rn) and⋂j∈Z

V j = 0,

(4) V0 is invariant under translations Tnn∈Zn ,

(5) If P0 is the orthogonal projection onto V0, then

ρT P0 = P0ρ, ∀ρ ∈ SO(n).

The next theorem gives conditions that are necessary for

D jaTkψl : k ∈Zn , j ∈N0, l = 1, . . . ,L∪ Tkφ : k ∈Zn

to be a tight frame for L2(Rn). It is cited from [38].

Theorem 5.5.14LET φ ∈ L2(Rn) be a refinable function with refinement mask H0 such that φ is continuousat the origin and lim|ξ|→0 φ(ξ) = 1. Furthermore for l = 1, . . . ,L let Hl ∈ L∞(T) be the waveletmasks for ψl ∈ L2(Rn).

THEN

D jaTkψl : k ∈Zn , j ∈N0, l = 1, . . . ,K ∪ Tkφ : k ∈Zn

is a tight frame for L2(Rn) with frame bound A = 1 if and only if for all q ∈D−1a Z

n/Zn andfor almost all ξ,ξ+q ∈T such that∑

k∈Zn

|φ(ξ+k)|2 > 0∧ ∑k∈Zn

|φ(ξ+q +k)|2 > 0,

the equalityL∑

l=0Hl (ξ)Hl (ξ+q) = δq,0 (5.9)

holds.

Proof. The proof can be found in [38] Theorem 4.7.

The tight wavelet frames defined via Corollary 5.5.6 and Theorem 5.5.7 satisfiy all conditionsbut condition (5.9) by construction.

Since supp(φ) =D−1a S, we have that

L∑l=0

H 2l (ξ) = H 2

0 (ξ)+H 21 (ξ) = 1, ∀ξ ∈ supp(φ) =D−1

a S

Hence it is sufficient for the conditions of Theorem 5.5.14 to be met, that


2 ·)

is a Lebesque zero set for any q ∈D−1a Z

n/Zn . It then follows that


2 ·) =;.

This is satisfied in the case of dyadic dilation Da = D2 as seen above.

5.6. IMPLEMENTATION AND APPLICATION 103

Real part Amplitude Phase Phase direction

Wav

elet

coef

fici

ents

Figure 5.5: Example of a monogenic wavelet decomposition. First row: Original image am-plitude, phase and phase direction of the monogenic signal. Second to fourth row: The firstthree scales of wavelet coefficients of the monogenic wavelet decomposition. Fifth row: Ap-proximation coefficients (low pass component). The first column displays the real part of thewavelet coefficients, i.e., the e0 component. The second and third column display the ampli-tude and the phase of the wavelet coefficients. The fourth column displays the color-codedphase direction: green points to the upper left and lower right corner, red encodes the hori-zontal direction, etc.. It is apparent that the phase direction displays the preferred directionson the corresponding scale. The descending sizes of the images represent the sub-samplingscheme of our filter-bank. The images show that the wavelet frame is extremely well adaptedfor round image features as well as for straight image features.

5.6 Implementation and application

The implementation of the monogenic wavelet has been the subject of the diploma thesis ofMartin Storath [47] under supervision of the author. A more detailed description of the imple-mentation can be found there.

5.6.1 Implementational aspects

The monogenic wavelet decomposition was implemented for 2-D and 3-D images as a plug-infor ImageJ [36]. For an example of the decomposition see Figure 5.5. The implementation usesthe isotropic wavelets constructed in Example 5.5.1 for the orders m = 0, . . . ,5 as given in (5.8)on page 95.


(a) Fourier filters h2,h1,h0. (b) Fourier filters r1h2,r1h1,r1h0.

Figure 5.6: Filter bank in the Fourier domain of the monogenic wavelets for the polynomialwith m = 3 and k = 2 sub-channels. (a) Fourier transform h of the real wavelet Ψ from left toright: First sub-channel (high pass filter), second sub-channel and low pass filter. (b) Fourier

transforms R1Ψ, i.e., (a) multiplied by the Fourier multiplier of the Riesz transform iξ1‖ξ‖ . Here

black ↔ -1, gray ↔ 0, and white ↔ 1.

The filters for the Riesz transform and the wavelet frames are infinite impulse response (IIR)filters. Therefore, a classical implementation in space domain based on convolution wouldtend to be rather costly. Furthermore there would be accuracy problems due to the necessityof truncating filters.

Since both the Riesz transform and the wavelet frames do have a simple and explicit form inthe Fourier domain, we implemented all algorithms in the frequency domain. The compactsupport of the wavelet frames in the Fourier domain allows a fast filter-bank implementationwith perfect reconstruction property. The Shannon sampling theorem ensures loss-less down-sampling after each filter step. As we constructed tight wavelet frames, the synthesis filter bankconsists of the same filters as the analysis filter bank.

The Fourier multiplier of the Riesz transform Rα(ξ) = iξα‖ξ‖ has a singularity at ξ= 0 (DC compo-

nent). We deal with this singularity by subtracting the images mean value beforehand, whichsets the DC component equal to zero.

For most applications, wavelet frames with dyadic dilations, as constructed above, are suffi-cient. Nevertheless, some applications deliver better results when a higher frequency resolu-tion is used resulting in better separation of phase components. (See Example 5.1.1.) For thatpurpose, we constructed wavelet frames with kp2-expansive dilations, k ∈N, i.e., we use thedilation matrix kp2In . From a signal processing point of view, the high-pass Fourier filter cor-responding to the dyadic case is divided into k sub-channels, which yields a (k +1)-channelfilter bank (k = 2 in Fig. 5.6). We chose the kp2 dilations because we are then able to reuse ourdyadic sub-sampling scheme after k filter steps.

The low-pass filter h0, the band-pass filters h j , j = 1, . . . ,k −1, and the high-pass filter hk arecomputed from the function h defined in Example 5.5.1 as follows:

high-pass: hk (ξ) : =

h(2k−1‖ξ|k ), if ‖ξ‖k 2k−1 < 1/4,

1, if ‖ξ‖k 2k−1 ≥ 1/4,

band-pass: h j (ξ) : = h(22k− j−1‖ξ‖k ),

low-pass: h0(ξ) : =

1, if ‖ξ‖k 22k−1 ≤ 1/4,

h(22k−1‖ξ‖k ), if ‖ξ‖k 22k−1 > 1/4,

where ξ ∈ [−1/2,1/2]n .

Figure 5.7 desribes the first step of the filterbank algorithm for dimension n = 2, two subchan-nels (k = 2) and an 2M ×2M -image f , where M ∈N. A general step of the algorithm looks as


f f h2 d1,2 h2 +++ f f

r1 h2 R1d1,2

r2 h2 R2d1,2

h1 d1,1 h1

r1 h1 R1d1,1

r2 h1 R2d1,1

h0 a1 h0

F F!1

F!1

F!1

F!1

F!1

F!1

F!1 ! 2

F!1

F!1

" 2 F!1

F

Figure 5.7: Filter bank for decomposition and reconstruction of signal a f by the monogenicwavelet frame. Here k = 2 high pass filters are used, namely h1 and h2. h0 is the approximationfilter. (See Fig. 5.6 (a) for the filter masks.) rl h j denotes the Fourier filter h j multiplied by the

Fourier multiplier of the Riesz transform rl = iξl‖ξ‖ (see Fig. 5.6 (b) for case l = 1). ds, j and Rl ds, j

are the real and hypercomplex detail coefficients, respectively. The first index s denotes thescale of the coefficients.

follows:

EXAMPLE 5.6.1 (A single step of the algotithm):Application of the wavelet filters hl of octave l = 1,2 in the Fourier domain yields the waveletcoefficients of the corresponding octave ds,l . Application of the Riesz transformed filters yieldsthe Riesz transformed wavelet octave filters R1ds,l ,R2ds,l . Application of the low pass filter h0

yields the approximation component as which is not Riesz transformed since it contains theDC-component. After downsampling the approximation component is the initial point forthe next step of the filter bank algorithm. The amplitude of the wavelet component As,l iscomputed as

As,l =√

d 2s,l +R1d 2

s,l +R2d 2s,l .

The phase φs,l is calculated as

φs,l = arg(ds,l + i

√R1d 2

s,l +R2d 2s,l

).

It follows thatds,l = As,l cos(φs,l ).

The phase direction Ds,l is computed via

Ds,l :=

arg

(R1ds,l√

R1d 2s,l+R2d 2

s,l

+ iR2ds,l√

R1d 2s,l+R2d 2

s,l

), if R2ds,l ≥ 0;

arg

(R1ds,l√

R1d 2s,l+R2d 2

s,l

+ iR2ds,l√

R1d 2s,l+R2d 2

s,l

)+π, else.

The runtime of the monogenic wavelet decomposition is determined by that of the fast Fouriertransform, namely O(N log N ). The memory consumption amounts to a factor of less than2(n +1) times the image size, where n = 2 or n = 3 is the image dimension. In the case of kp2-expansive dilations, the runtime and the redundancy increase by a factor of k as compared tothe dyadic case. E.g., for the application “Equalization of Brightness” (see section 5.6.2), k = 5turned out to be a sensible trade-off between frequency resolution and runtime.

We computed images of different sizes with the ImageJ implementation of our fast algorithm.The runtime and accuracy results of the experiments are given in Table 5.1. The examples were


Table 5.1: Computation time and accuracy of the Riesz wavelet filter bank for various samplesizes. The time for the creation of the filter-bank is denoted as pre-calculation time. The re-construction error lies in the range of machine accuracy. The Examples were computed on anIntel Core 2 Duo with 2.33 GHz.

Sample Size Precalculation Analysis Synthesis l∞-Reconstruction Equalization

Time of brightness

in sec in sec in sec Error in sec128 × 128 0.02 0.03 0.05 4.2 ·10−16 0.19256 × 256 0.10 0.10 0.14 5.1 ·10−16 1.63512 × 512 0.40 0.45 0.56 6.7 ·10−16 6.66

1024 × 1024 2.00 1.96 2.36 9.1 ·10−16 28.67128×128×128 3.45 2.66 4.20 3.9 ·10−16 71.43

calculated using dyadic dilations. In section 5.6.2 below, we used 5p2-expansive dilations. Notethat the filters have to be computed only once because they are used both for decompositionand reconstruction.

5.6.2 Applications

Equalization of Brightness

Looking at the monogenic wavelet decomposition we, observe that the amplitude containsinformation about the brightness of the image, whereas the phase contains the structure ofthe image. Note that this observation is true only for the amplitude and phase of the waveletcoefficients of the image, and not for the amplitude and phase decomposition of the imageitself. (Compare Fig. 5.11.)

Hence, reconstruction by phase only delivers an image with balanced brightness (Fig. 5.8).This reconstruction by phase only is achieved by setting the amplitude of the wavelet coeffi-cients to be constant. Details are conserved or even enhanced by equalization of brightness.As mentioned above the results become better when using kp2-expansive dilations instead ofdyadic ones (we chose k = 5). The algorithm also sets very low amplitudes to one resultingin an amplification of noise. Therefore, we added a simple noise suppression based on themagnitude of the amplitude.

Equalization of brightness can also be applied to color images. See [47] for the details.

The multi-scale phase concept is closely connected to the human visual perception. This canbe seen in the example of monogenic bandpass filters [16] applied to Adelsons checkerboardillusion [1]. In Figure 5.10, we reproduce this effect using our monogenic wavelet frame. Notethat in contrast to [16] no structural information is lost since we use a wavelet frame insteadof a set of bandpass filters. Hence the resulting images are far closer to the original images.(Compare Figure 5.10 with figure 1 in [16].)

Another hint that the human visual perception is related to a multi-scale phase concept is pro-vided by the optical illusion in Figure 5.9. The image consists of black squares separated bywhite stripes, but the human vision injects small gray disks into the intersection of the hor-izontal and vertical stripes. This phenomenon is reproduced in the phase image derived byequalization of brightness.


(a) Corrupted Image (b) Equalization of Brightness

Figure 5.8: (a) Artificially corrupted image. Gray levels are multiplied by a ramp function and by 0.05 in the center.(b) Reconstruction from phase only. The brightness is now balanced, in particular the difference of brightness betweenthe dark center and its outside disappears. Even fine image structures are recovered. Decomposition was computedusing k = 5 sub-channels.

Parameter Free Descreening

An application using the amplitude of the monogenic wavelet coefficients is parameter freedescreening. Screening effects in images are usually removed by applying an adequate lowpass filter. The filter size has to be given as a parameter or estimated in some way. We usethe fact that due to the regularity of the screening effect the maximal amplitude lies in thesubband corresponding to the screening effect to automatically find the adequate low passfilter that removes the screening [24, 50].


(a) Image with optical illusion (b) Equalization of brightness

Figure 5.9: The optical illusion of injecting gray disks is reproduced by the phase image (b)derived by Equalization of Brightness.

(a) Original (b) Phase (c) Reconstruction from Phase

Figure 5.10: (a) Adelsons checkerboard illusion [1]: Squares A and B seem to have differentbrightness, but in fact they have the same gray levels. (b) In the phase image, A and B havedifferent gray levels. The contour of the shadow is still visible. (c) After the reconstruction fromphase of wavelet scales, A and B have again different gray levels and the contour of the shadowdisappeared. Regarding the structure of the image, the human vision is in some way relatedto image (c): The absolute gray values in image (a) do not yield a checkerboard structure, asA and B have the same gray values. But we recognize (a) in the way of (c): Our visual systemcompensates the shadow that is thrown by the cylinder. (Compare also [16].)


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Chapter 6

Uncertainty relations for the Riesztransform

Uncertainty relations quantify the joint localisation in time, or space domain, and Fourier do-main. For that reason they are an important tool in the design of wavelets. In chapter 5 we im-plement Hilbert and Riesz transform via wavelets. Therefore it will be the topic of this chapterto compute the effect of the Hilbert and Riesz transforms on the Weyl-Heisenberg uncertaintyrelation.

In Theorem 6.2.3 we proof the new result that under mild assumptions the Weyl-Heisenberguncertainty relation is invariant under the Hilbert transform.

Since the Riesz transform R is unitary (See Theorem 4.1.4.) whereas the partial Riesz trans-forms Rα are not it seems reasonable to consider a single uncertainty relation for the Riesztransform rather than a set of uncertainty relations for the partial Riesz transforms. This re-quires a new kind of uncertainty relation for vector valued functions.

We pursue this goal along two different paths: A classical approach Theorem 6.3.2 based onthe Cauchy-Schwartz inequality and an uncertainty relation on Clifford-Hilbert modules The-orem 6.3.3. Both paths yield novel uncertainty relations.

In contrast to the set of classical Weyl-Heisenberg uncertainty relations in higher dimensionsstated in Corollary 6.3.1 the Weyl-Heisenberg uncertainty relation we derive in Theorem 6.3.10is invariant under rotations. This property of invariance with respect to rotation is very desir-able for image processing.

In Theorem 6.3.12 we use our results to derive a novel Weyl-Heisenberg uncertainty relation forthe Riesz transform and to compute the effect of the Riesz transform on the Weyl-Heisenberguncertainty relation stated in Theorem 6.3.10.

Finally we state a new affine uncertainty relation Theorem 6.3.13 as a further example for anapplication of Theorem 6.3.3.

Part of the results of this chapter have been published in the peer reviewed article [22].

111

112 CHAPTER 6. UNCERTAINTY RELATIONS

6.1 The Heisenberg algebra and the Weyl Heisenberg group

Definition 6.1.1 (Heisenberg algebra and Weyl-Heisenberg group)LET a,b,c,d ∈Rn and s, t ∈R.

THEN (a,b, t ), (c,d , s) ∈R2n+1 and we define a Lie bracket onR2n+1 by

[(a,b, t ), (c,d , s)] = (0,0, ad −bc).

The Lie algebra hn := (R2n+1, [ · , · ]) is called the Heisenberg algebra. (Lie algebras and Liegroups are defined in the Appendix, subsection A.1.1.)

If A1, . . . , An ,B1, . . . ,Bn ,T is the standard basis of R2n+1 then

[A j , Ak ] = [B j ,Bk ] = [A j ,T ] = [B j ,T ] = 0, and [A j ,Bk ] = δ j ,k T.

A representation of the Heisenberg algebra on S (Rn) is given by

dWa,b,t f (x) := (a ·∇+2πi b · x +2πi t Id) f (x), ∀ f ∈S (Rn), x ∈Rn .

That is the mapping from hn to the set of skew-Hermitian operators dWa,b,t a,b∈Rn , t∈R is a Liealgebra homomorphism.

The Weyl-Heisenberg group Hn is R2n+1 with group law

(a,b, t )(c,d , s) = (a + c,b +d , t + s +1/2(ad −bc)).

The Weyl-Heisenberg group is the Lie group corresponding to the Heisenberg Lie algebra.

The Schrödinger representation of Hn is given by the set of unitary operatorsWa,b,t a,b∈Rn , t∈R, where

Wa,b,t : L2(R) → L2(R), f 7→ e2πi (t+ab)MbTa f .

The infinitesimal operators of a group representation are the derivatives at the identity of theoperator defining the representation with respect to the basis elements of the group represented.The infinitesimal operators of a Lie group are a basis for a representation of the correspondingLie algebra.

Proposition 6.1.2 (Infinitesimal operators of the Weyl-Heisenberg group)The infinitesimal operators of the Weyl-Heisenberg group are

tα f (x) := d

d aαe2πi (t+ab)MbTa f (x)

∣∣a=b=0, t=0

= 2πi bαe2πi (t+ab)MbTa f (x)−e2πi (t+ab)MbTa∂

∂xαf (x)

∣∣a=b=0, t=0

=− ∂

∂xαf (x),

sα f (x) := d

dbαe2πi (t+ab)MbTa f (x)

∣∣a=b=0, t=0

= 2πi aαe2πi (t+ab)MbTa f (x)+2πi xαe2πi (t+ab)MbTa f (x)∣∣

a=b=0, t=0

= 2πi xα f (x),

d

d te2πi (t+ab)MbTa f

∣∣a=b=0, t=0 = 2πi f .

As we have noted above these operators are a basis for the representation of the Heisenbergalgebra on S (Rn).

6.2. UNCERTAINTY RELATIONS IN ONE DIMENSION 113

6.2 Uncertainty relations in one dimension

Uncertainty principles in one dimension are based on a uncertainty relation for self-adjointoperators stated in this form for example in [12].

Theorem 6.2.1LET H be a Hilbert space and let S,T be densely defined self-adjoint operators

S : H ⊇ Dom(S) → H ,

T : H ⊇ Dom(T ) → H .

Furthermore, let the commutator [S,T ] be given by

[S,T ] := ST −T S.

THEN for any function f ∈ H and any pair of real numbers s, t ∈ R the following inequalityholds: ∣∣⟨[S,T ] f , f ⟩∣∣2 ≤ 4‖(S − s) f ‖2‖(T − t ) f ‖2. (6.1)

Equality in (6.1) holds iff∃λ ∈R : (S − s) f = iλ(T − t ) f .

If ‖ f ‖ = 1, then equality in (6.1) implies that

s = ⟨S f , f ⟩, and t = ⟨T f , f ⟩.

Furthermore (6.1) can be rewritten as∣∣⟨[S,T ] f , f ⟩∣∣2 ≤ 4⟨(T − t )2 f , f ⟩⟨(S − s)2 f , f ⟩

⟨(T − t )2 f , f ⟩ is called the variance of T .


The operators Sα = i sα and Tβ = i tβ in Proposition 6.1.2 are self-adjoint and have a non-vanishing commutator iff α = β. Hence we can derive an uncertainty relation of them viaTheorem 6.2.1 - the Heisenberg uncertainty in one dimension. (See for example [20] whichcontains a beautiful introduction to uncertainty relations.)

Theorem 6.2.2 (Heisenberg uncertainty relation)LET f ∈ L2(R), and let l ,r ∈R.

THEN

16π2∫R

(x + r )2| f (x)|2d x∫R

(ξ+ l )2| f (ξ)|2dξ≥ ‖ f ‖4. (6.2)

Equality holds if f (x) = e−πx2

λ , for some λ ∈R.

Proof. Let S f (x) := −2πx f (x) and T f (x) := −i f ′(x). S and T are densely defined self-adjointoperators on L2(R). Furthermore let s = 2πr and t = 2πl . We can apply (6.2.1) which statesthat

4‖(S − s)2 f ‖2‖(T − t )2 f , f ‖2 ≥ ∣∣⟨[S − s,T − t ] f , f⟩∣∣2.


Then

[S − s,T − t ] = (S − s)(T − t )− (T − t )(S − s)

= ST −T S − sT +Ts −St + tS + st − t s

= [S,T ].

Hence, the right hand side of the inequality (6.2.1) yields

∣∣⟨[(S − s), (T − t )] f , f ⟩∣∣2 =∣∣∣∫R

2πi x f ′(x)−2πi∂

∂x(x f (x)) f (x)d x

∣∣∣2

=∣∣∣∫R

2πi(x f ′(x)− f (x)−x f ′(x)

)f (x)d x

∣∣∣2

= ∣∣2πi∣∣2‖ f ‖4.

The first term of the left hand side of (6.2.1) is simply

⟨(S − s)2 f , f ⟩ =∫

(−2πx − s)2 f (x) f (x)d x =∫

(−2πx − s)2| f (x)|2d x.

The second term gives

⟨(T − t )2 f , f ⟩ = ⟨(T − t ) f , (T − t ) f ⟩ = ‖(T − t ) f ‖2 = ∥∥F((T − t ) f

)∥∥2

=∫R

(−2πx − t ) f (x)(−2πx − t ) f (x)d x

=∫R

(−2πx − t )2| f (x)|2d x.

Equivalence holds if ∃λ ∈R : (S − s) f = iλ(T − t ) f . This is clearly true if f (x) = e−πx2

λ .

6.2.1 The Heisenberg uncertainty and the Hilbert transform

The following new theorem states that the Heisenberg uncertainty relation introduced in The-orem 6.2.2 is invariant under the Hilbert transform if the first moment vanishes.

Theorem 6.2.3 (Heisenberg uncertainty relation and the Hilbert transform)LET f ∈ L2(R,R) such that S f ,T f ∈ L2(R) and additionally

∫R

f (x)d x = 0.

THEN ∫R

x2| f (x)|2d x∫R

ξ2| f (ξ)|2dξ=∫R

x2|H f (x)|2d x∫R

ξ2|H f (ξ)|2dξ.

Proof. We first show that SH f ∈ L2(R). Indeed we show that T H f ∈ L2(R). First note that ifS f ∈ L2(R) then f is an element of the Sobolev space

W 1,2(R) := g ∈ L2(R) : g ′ ∈ L2(R)

,

with norm ‖.‖W 1,2 given by ‖g‖W 1,2 = ‖g‖L2(R) +‖g ′‖L2(R), ∀g ∈ W 1,2(R). By the Sobolev em-

bedding theorem (See [26] theorem 4.5.11) f ∈C (R) and consequently f (0) = 0.

Let fk k∈N ⊂C∞(R) such that fk (0) = 0 and

fkk→∞−→ f in W 1,2(R).

6.3. UNCERTAINTY IN HIGHER DIMENSIONS 115

In the distributional sense sgn′ = 2δ0 and hence

( fk sgn)′(φ) = f ′k sgn(φ)+2 fkδ0(φ)

=∫R

f ′k (x)sgn(x)φ(x)d x +2 fk (0)φ(0)︸︷︷︸

=0

, ∀φ ∈D(R).

It follows from the Cauchy-Schwarz inequality that∣∣ f ′k sgn(φ)− f ′ sgn(φ)

∣∣= ∣∣∫

R

(f ′

k (x)sgn(x)− f ′(x)sgn(x))φ(x)d x

∣∣≤

√∫R

∣∣( f ′k (x)− f ′(x)

)sgn(x)

∣∣2d x

√∫Rn

|φ(x)|2d x

k→∞−→ 0, ∀φ ∈D(R)

This shows that H f′ = i sgn f ′ ∈ L2(R). (6.3)

We can now compute∫R

x2|H f (x)|2d x∫R

ξ2|H f (ξ)|2dξ

=∫R

1

4π2

∣∣∣ ∂∂ξ

àH f (ξ)∣∣∣2

dξ∫R

ξ2|i sgn(ξ) f (ξ)|2dξ

(6.3)= 1

4π2

(∫R+

∣∣i sgn(ξ) f ′(ξ)∣∣2dξ+

∫R−

∣∣i sgn(ξ) f ′(ξ)∣∣2dξ

)∫R

ξ2| f (ξ)|2dξ

= 1

4π2

∫R

∣∣ f ′(ξ)∣∣2dξ

∫R

ξ2| f (ξ)|2dξ

=∫R

x2| f (x)|2d x∫R

ξ2| f (ξ)|2dξ.

6.3 Uncertainty in higher dimensions

We have just seen that the Hilbert transform does not change the localization of a function,i.e., the Heisenberg uncertainty relation is invariant under the Hilbert transform. If we wantto know if this also applies to the Riesz transform, we have to compute the Heisenberg uncer-tainty of the Riesz transform of a function and compare it to the Heisenberg uncertainty of theoriginal function. Therefore we derive an uncertainty relation for vector valued functions likethe Riesz transform of a function.

The uncertainty principle Theorem 6.2.2 is usually generalized to higher dimensions by apply-ing Theorem 6.2.1 to the partial infinitesimal operators calculated in Proposition 6.1.2. (See forexample [17] Corollary 2.6.)

Corollary 6.3.1LET f ∈ L2(Rn), and l = (l1, . . . , ln),r = (r1, . . . ,rn) ∈Rn .

THEN

16π2∫Rn

(xα+ rα)2| f (x)|2d x∫Rn

(ξα+ lα)2| f (ξ)|2dξ≥ ‖ f ‖4, ∀α= 1, . . . ,n. (6.4)


Corollary 6.3.1 states a set of n uncertainty relations. We suggest two ways to get a single un-certainty relation.

In Theorem 6.3.2, we first derive a uncertainty relation from Corollary 6.3.1.

In Theorem 6.3.3 we consider the infinitesimal operators derived from the n-dimensionalWeyl-Heisenberg group by application of Dirac operators of the form

∑nα=1 eα

∂∂xα

to obtain

a single uncertainty relation, rather than partial derivatives of the form i ∂∂xα

to get n uncer-tainty relations. This results in an uncertainty relation for operators in Clifford-Hilbert mod-ules which is more general but also a lot weaker than that in Theorem 6.3.2.

6.3.1 A single uncertainty relation for sets of self adjoint operators

Theorem 6.3.2LET Sα,Tα, α= 1, . . . ,n, be densely defined self-adjoint operators

Sα : H ⊇ Dom(Sα) → H ,

Tα : H ⊇ Dom(Tα) → H ,

such that ∩αDom(Sα) and ∩αDom(Tα) are dense in H .

THEN for any set of functions f1, . . . , fm ∈ ∩nα=1 Dom([Sα,Tα]) and any pair of vectors

(s1, . . . , sn), (t1, . . . , tn) ∈Rn the following inequality holds:

( n∑α=1

m∑k=1

∣∣⟨[Sα,Tα] fk , fk⟩∣∣)2 ≤ 4

n∑α=1

m∑k=1

‖(Sα− sα) fk‖2n∑α=1

m∑k=1

‖(Tα− tα) fk‖2. (6.5)

Equality in (6.5) holds iff

∃λ ∈R : (Sα− sα) fk = iλ(Tα− tα) fk , ∀α= 1, . . . ,n, k = 1, . . . ,m.

If ‖ fk‖ = 1, then equality in (6.5) implies that

sα = ⟨Sα fk , fk⟩, and tα = ⟨Tα fk , fk⟩.

Proof. The idea to use the Cauchy-Schwartz inequality to combine a set of uncertainty rela-tions into a single one is inspired by a proof of a Heisenberg uncertainty relation in n-D whichcan be found in [17] Corollary 2.8 see Remark 6.3.11 for details.

We now use this idea to proof the uncertainty relation for sets of self-adjoint operators appliedto vector valued functions.

By theorem Theorem 6.2.1 we know that∣∣⟨[Sα,Tα] fk , fk⟩∣∣≤ 2‖(Sα− sα) fk‖‖(Tα− tα) fk‖, ∀α= 1, . . . ,n; k = 1, . . . ,m.

Summation over α and k and application of the Cauchy-Schwarz inequality forRn yields

n∑α=1

m∑k=1

∣∣⟨[Sα,Tα] fk , fk⟩∥∥≤ 2

n∑α=1

n∑k=1

‖(Sα− sα) fk‖‖(Tα− tα) fk |

≤ 2

√n∑α=1

m∑k=1

‖(Sα− sα) fk‖2

√n∑α=1

m∑k=1

‖(Tα− tα) fk‖2.


Equality holds iff

∃λ ∈ iR : (Sα− sα) fk =λ(Tα− tα) fk , ∀α= 1, . . . ,n; k = 1, . . . ,m.

In this case it follows that

⟨(Sα− sα) fk , fk⟩ =λ⟨(Tα− tα) fk , fk⟩, ∀α= 1, . . . ,n; k = 1, . . . ,m,

and hence λ= 0.

Let ‖ fk‖ = 1. Then sα = ⟨Sα fk , fk⟩ and by the same argument tα = ⟨Tα fk , fk⟩.

6.3.2 Uncertainty relations in Clifford-Hilbert modules

The infinitesimal operators of the Heisenberg group computed in Proposition 6.1.2 are not selfadjoint. To apply the uncertainty relation we multiplied by i to get self adjoint operators. In aClifford algebra setting we interpret the basis vectors as hypercomplex elements of an algebra,hence the operators S =∑

α eαsα and T =∑α eαtα are self adjoint.

To apply an uncertainty relation to these operators we have to extend Theorem 6.2.1 to thesetting of Clifford-Hilbert modules which we introduced in chapter 3.2. By this approach wewill state a novel uncertainty relation for Clifford-Hilbert modules.

Theorem 6.3.3 (Uncertainty relation in Cliffor-Hilbert modules)LET H be a complex Hilbert space. Furthermore let S,T be densely defined self-adjoint left-linear operators on Hn .

S : Hn ⊇ Dom(S) → Hn

T : Hn ⊇ Dom(T ) → Hn .

The commutator [S,T ] is given by

[S,T ] := ST −T S.

THEN

(i) ∣∣([S,T ] f , f)∣∣2 ≤ 4‖S f ‖2‖T f ‖2. (6.6)

(ii) Equality in (6.6) holds iff

∀α ∈On ∃λ ∈R+0 : ⟨S f ⟩α = iλ⟨T f ⟩α. (6.7)

And furthermore

‖⟨T f ⟩α‖‖⟨S f ⟩β‖ = ‖⟨T f ⟩β‖‖⟨S f ⟩α‖,∀α,β ∈On . (6.8)

( Recall (3.1): On := α ∈P(1, . . . ,n) : 1 ≤α1 <α2 < . . . <αν ≤ n; ν ∈ 1, . . . ,n∪ 0

. )


Proof. ad(i)

([S,T ] f , f ) = (T f ,S f )− (S f ,T f )

= ∑α∈On

⟨⟨T f ⟩α,⟨S f ⟩α

⟩H−

⟨⟨S f ⟩α,⟨T f ⟩α

⟩H

=2∑

α∈On

ℑ(⟨⟨T f ⟩α,⟨S f ⟩α⟩).

Hence

|([S,T ] f , f )|2 =(2∣∣ ∑α∈On

ℑ(⟨⟨T f ⟩α,⟨S f ⟩α⟩)∣∣)2

≤4( ∑α∈On

∣∣ℑ(⟨⟨T f ⟩α,⟨S f ⟩α⟩)∣∣)( ∑

β∈On

∣∣ℑ(⟨⟨T f ⟩β,⟨S f ⟩β⟩)∣∣) (6.9)

≤4∑

α,β∈On

‖⟨T f ⟩α‖‖⟨S f ⟩α‖‖⟨T f ⟩β‖‖⟨S f ⟩β‖ (6.10)

=4(2

∑α<β∈On

‖⟨T f ⟩α‖‖⟨S f ⟩β‖‖⟨T f ⟩β‖‖⟨S f ⟩α‖

+ ∑α∈On

‖⟨T f ⟩α‖2‖⟨S f ⟩α‖2)

(6.11)

≤4∑

α,β∈On

‖⟨T f ⟩α‖2‖⟨S f ⟩β‖2 (6.12)

=4‖T f ‖2‖S f ‖2,

where (6.10) holds by the Cauchy-Schwartz inequality and (6.12) holds by the inequality2ab ≤ a2+b2, ∀a,b ∈R. The order of On in (6.11) may be chosen arbitrarily, an examplewould be the lexicographical order.

ad(ii) For equality in (6.6) to hold we require equality in (6.10), (6.11) and (6.12).

(6.10) holds with equality iff ∀α ∈ On ∃λ ∈ R : ⟨(S − s) f ⟩α = iλ⟨(T − t ) f ⟩α. Now (6.9)holds with equality iff λ≥ 0.

Finally (6.12) holds with equality, iff (6.8) is fulfilled.

Remark 6.3.4The right hand sides of Theorem 6.3.2,

4n∑

j=1

m∑k=1

‖(S j − s j ) fk‖2n∑

j=1

m∑k=1

‖(T j − t j ) fk‖2

and Theorem 6.3.3,4‖S f ‖2‖T f ‖2 = ∑

α∈On

‖⟨S f ⟩α‖2∑β∈On

‖⟨T f ⟩β‖2

coincide if the vector part of the operators S,T or the function f in Theorem 6.3.3 vanishes.

If the vector part of the operator vanishes, then

⟨S f ⟩α = S0 fα

which, for n = 1, yields Theorem 6.3.2.


If the vector part of the function f vanishes, then

⟨S f ⟩α = Sα f0

which, for m = 1, yields Theorem 6.3.2.

The most important point in which the left hand sides differ is that the left hand side of The-

orem 6.3.2(∑n

j=1

∑mk=1

∣∣⟨[S j ,T j ] fk , fk⟩∣∣)2

is translation invariant, while the left hand side of

Theorem 6.3.3∣∣([S,T ] f , f

)∣∣2 is not. This can be amended with additional prerequisites - this isthe topic of the following section.

Translation invariant uncertainty relations

In the following three corollaries we will address the problem that in general

[S − s,T − t ] 6= [S,T ],

that is, the uncertainty principle Theorem 6.3.3 is not translation invariant.

Corollary 6.3.8 and Theorem 6.3.3 are special cases of Theorem 6.3.2, for n = 1, respectively,m = 1. Corollary 6.3.6 is a special case of Theorem 6.3.2 if Sα fβ = Sβ fα and Tα fβ = Tβ fα,∀α,β ∈ 1, . . . ,n.

This section shows the difficulties in deriving a translation invariant version of Theorem 6.3.3– whenever applicable Theorem 6.3.2 is stronger and easier to use than Theorem 6.3.3. Theproper domain of application for Theorem 6.3.3 would be operators S,T and functions f whichuse the full range of indices On .

Remark 6.3.5 (Linear operators in Clifford-Hilbert modules)We have seen in Theorem 3.2.7 (viii) that an operator O on the Clifford-Hilbert module has theform

O = ∑α∈On

eαOα,

where Oα : H → H .

Note that in order to be left-linear the operators are always applied from the right, i.e.,

O f = ∑α,β∈On

eαeβOβ( fα).

This means that the linear operator c Id is a multiplication by c from the right rather than theaction of Cn on HN which is multiplication from the left by c.

If O is not defined on the whole Hilbert-Clifford module denote

Dom(O) = ⋂α∈On

Dom(Oα)

andDomn(O) := ∑

α∈On

eαDom(O).

O f is well defined for some f ∈ L2n(Rn) iff f ∈ Domn(O), i.e.

fα ∈ Dom(O), ∀α ∈On .


Corollary 6.3.6LET S,T be as in Theorem 6.3.3 and such that S =∑n

α=1 eαSα and T =∑nα=1 eαTα. Furthermore,

let s, t ∈Rn ⊂Rn be two vectors and let f =∑nα=1 eα fα ∈ F ⊆ Hn ∩Dom([S,T ]). Finally, let for

all α,β ∈ 1, . . . ,n, ⟨((Sα− sα)(Tβ− tβ)+ (Tβ− tβ)(Sα− sα)

)fα, fβ

⟩(6.13)

=⟨(

(Sα− sα)(Tβ− tβ)+ (Tβ− tβ)(Sα− sα))

fβ, fα⟩

.

THEN ∣∣([S − s,T − t ] f , f )∣∣= ∣∣ n∑

α,β=1⟨[Sα,Tα] fβ, fβ⟩

∣∣,and hence ∣∣ n∑

α,β=1⟨[Sα,Tα] fβ, fβ⟩

∣∣2 ≤ 4‖(S − s) f ‖2‖(T − t ) f ‖2.

Remark 6.3.7(6.13) is a technical condition deduced from the calculation given in the following proof. It isfulfilled for example if fα = fβ, ∀α,β= 1, . . . ,n.

Proof.

([S − s,T − t ] f , f ) = ∑α,β,γ,δ=1,...,n:

eγeαeβ=±eδ

⟨(Sα− sα)(Tβ− tβ) fγ, fδ⟩eγeδeβeα

−⟨(Tβ− tβ)(Sα− sα) fγ fδ⟩eγeδeαeβ

=n∑

α,β=1⟨[Sα− sα,Tα− tα] fβ, fβ⟩eβeβeαeα︸︷︷︸

=−1

+ ∑α,β=1,...,n

α6=β

(⟨((Sα− sα)(Tβ− tβ)+ (Tβ− tβ)(Sα− sα)

)fα, fβ

⟩eαeβeβeα︸︷︷︸

=−1

+⟨(

(Sα− sα)(Tβ− tβ)+ (Tβ− tβ)(Sα− sα))

fβ, fα⟩

eβeαeβeα︸︷︷︸=1

)

=−n∑

α,β=1⟨[Sα− sα,Tα− tα] fβ, fβ⟩

=−n∑

α,β=1⟨[Sα,Tα] fβ, fβ⟩.

Corollary 6.3.8LET S,T be densely defined self adjoint operators

S : Hn ⊇ Domn(S) → Hn ,

T : Hn ⊇ Domn(T ) → Hn .

Let f = f0e0 ∈ Hn , where f0 ∈ H . Furthermore let s, t ∈Cn be a pair of Clifford algebra valuednumbers such that s Id and t Id are self adjoint, as in Example 3.2.2 on page 57.

THEN


(i) ∣∣([S,T ] f , f)∣∣2 = ∣∣ ∑

α∈On

⟨[Sα,Tα] f0, f0⟩∣∣2 ≤ 4‖(S − s) f ‖2‖(T − t ) f ‖2. (6.14)


∃λ ∈R+0 : (S − s) f = iλ(T − t ) f , (6.15)

and furthermore

‖(T − t )α f0‖‖(S − s)β f0‖ = ‖(T − t )β f0‖‖(S − s)α f0‖, ∀α,β ∈On . (6.16)

(iii) If ‖ f ‖ = 1, then equality in (6.14) implies that

s =n∑

α∈On

eα⟨Sα f0, f0⟩, and t = ∑α∈On

eα⟨Tα f0, f0⟩.

Proof. Let S = ∑α∈On

eαSα, T = ∑α∈On

eαTα, where Sα : Dom(S) → H and Tα : Dom(S) → H .Furthermore let s =∑

α∈Oneαsα, t =∑

α∈Oneαtα, where sα, tα ∈C.

ad(i) We only need to show that([(S − s), (T − t )] f , f

) = ([S,T ] f , f

). The rest follows by Theo-

rem 6.3.3(i).

Let s, t ∈Cn . Then

[(S − s), (T − t )] = (ST −T S)− (sT −Ts)− (St − tS)+ (st − t s).

This is in general certainly non zero, since the Clifford algebra is not commutative. Letus consider (

[S − s,T − t ] f , f)= ⟨⟨[(S − s), (T − t )]⟩0 f0, f0

⟩H

= ⟨⟨(ST −T S)− (sT −Ts)− (St − tS)+ (st − t s)⟩0 f0, f0⟩

H .

Hence, if the identity⟨(st − t s)− (sT −Ts)− (St − tS)⟩0 = 0 (6.17)

holds, it is true that([S − s,T − t ] f , f

)= ([S,T ] f , f

), ∀ f ∈ Hn : fα = 0, ∀α 6= 0.

Indeed, since sβ, tβ ∈C and Sβ,Tβ ∈ H , ∀β ∈On ,

⟨st − t s⟩0 =n∑β=1

e2β(sβtβ− tβsβ) = 0,

⟨St − tS⟩0 =n∑β=1

e2β(Sβtβ− tβSβ) = 0,

and finally

⟨sT −Ts⟩0 =n∑β=1

e2β(sβTβ−Tβsβ) = 0.

Now (S − s) and (T − t ) satisfy the conditions for Theorem 6.3.3 which proves (i).


ad(ii) See Theorem 6.3.3(ii).

ad(iii) First note that⟨⟨

S f , f⟩⟩

Id and⟨⟨

T f , f⟩⟩

Id are self adjoint. For this note that by Propo-sition 3.2.10(c) ⟨Sα f0, f0⟩ ∈R, ∀α : eα = eα and ⟨Sα f0, f0⟩ ∈ iR ∀α : eα = eα. But this

means that⟨⟨

S f , f⟩⟩= ⟨⟨

S f , f⟩⟩

and hence that⟨⟨

S f , f⟩⟩

Id is self adjoint.

Now if we take the inner product of both sides of (6.15) we get⟨⟨(S − s) f , f

⟩⟩= iλ⟨⟨

(T − t ) f , f⟩⟩

⇔ ∑α∈On

eα⟨⟨S − s⟩α f0, f0⟩ = iλ∑

α∈On

eα⟨⟨S − s⟩α f0, f0⟩

⇔ ⟨⟨S − s⟩α f0, f0⟩ = iλ⟨⟨T − t⟩α f0, f0⟩, ∀α ∈On

⇔⟨Sα f0, f0⟩ = ⟨s⟩α and ⟨Tα f0, f0⟩ = ⟨t⟩α, ∀α ∈On

⇔ s = ⟨⟨S f , f

⟩⟩and t = ⟨⟨

T f , f⟩⟩

.

Corollary 6.3.9LET S0,T0 be densely defined self adjoint operators

S : H ⊇ Dom(S) → H

T : H ⊇ Dom(T ) → H ,

and let S = e0S0, T = e0T0 : Hn → Hn . Let f ∈ Hn , and let s, t ∈R.

THEN

(i) ∣∣([S,T ] f , f)∣∣2 = ∣∣ n∑

α=1⟨[S0,T0] fα, fα⟩

∣∣2 ≤ 4‖(S − s) f ‖2‖(T − t ) f ‖2. (6.18)


∀α ∈On ∃λ ∈R+0 : (S0 − s0) fα = iλ(T0 − t0) fα. (6.19)

And furthermore for all α,β ∈On

‖(T0 − t0) fα‖‖(S − s) fβ‖ = ‖(T − t ) fβ‖‖(S − s) fα‖ (6.20)

(iii) If ‖ fα‖ = 1, ∀α ∈ O ⊆On , and fα = 0; ∀α ∉ O, then equality in (6.18) implies that for allα ∈O

s0 = ⟨S0 fα, fα⟩ and t0 = ⟨T0 fα, fα⟩.

Proof.

ad(i) We only need to show that([(S − s), (T − t )] f , f

) = ([S,T ] f , f

). The rest follows by Theo-

rem 6.3.3(i).

Indeed

[S − s,T − t ] = ST −T S − sT +Ts −St + tS + st − t s

= e0(S0T0 −T0S0 − s0T0 +T0s0 −S0t0 + t0S0 + s0t0 − t0s0

)= e0(S0T0 −T0S0)

= [S,T ].


ad(ii) See Theorem 6.3.3(ii).

ad(iii) Let ‖ f ‖ = 1. Then equality in (6.18) implies (6.19) and hence that

∃λ ∈R+0 : (S0 − s0) fα = iλ(T0 − t0) fα, ∀α ∈On .

It follows that

⟨(S0 − s0) fα, fα⟩ = iλ⟨(T0 − t0) fα, fα⟩.

Since (S0 − s0) and (T0 − t0) are self adjoint operators, we know that λ = 0 and for allα ∈On

s0⟨ fα, fα⟩ = ⟨S0 fα, fα⟩t0⟨ fα, fα⟩ = ⟨T0 fα, fα⟩.

6.3.3 The Weyl-Heisenberg uncertainty

Applying Corollary 6.3.8 to the infinitesimal operators derived from the n-dimensional Weyl-Heisenberg group Definition 6.1.1 by the Dirac operator D =∑n

α=1 eα∂∂xα

leads to the isotropicuncertainty relation stated below. This theorem is already known ([17] Corollary 2.8). However,we will give a new proof. This proof is based on the fact that the operator D is the square rootof the Laplace operator.

While we are not in a setting where applying Theorem 6.3.3 via Corollary 6.3.8 is strictly nec-essary, the new proof shows the concepts behind the Weyl-Heisenberg uncertainty relation –vector valued infinitesimal operators, which are square roots of the modulus respectively theLaplace operator.

Theorem 6.3.10 (A single Weyl-Heisenberg uncertainty relation forRn)LET f0 ∈ L2(Rn), l ,r ∈Rn .

THEN the following inequality holds:

16π2∫Rn

|x + l |2| f (x)|2d x∫Rn

|x + r |2| f (x)|2d x ≥ n2‖ f ‖4.

Equality holds if f (x) = e−πλ|x|2 , for some λ ∈R+.

Proof. Let s = ∑nα=1 eαsα = 2πi

∑nα=1 eαrα and t = ∑n

α=1 eαtα = 2πi∑nα=1 eαlα. Note that s Id

and t Id are self-adjoint left linear operators on L2n(Rn). (See Example 3.2.2.)

In order to proof the theorem we apply inequality (6.14), which states that

4‖(S − s) f ‖2‖(T − t ) f ‖2 ≥ ∣∣([S,T ] f , f)∣∣2,

to the function f = e0 f0 and the operators given for suitable functions in g ∈ L2(Rn)n by

Sg (x) = 2πi∑β∈On

n∑α=1

eβeαxαgβ(x)


and

T g (x) = Dg (x) = ∑β∈On

n∑α=1

eβeα∂

∂xαgβ(x),

respectively.

The right hand side of (6.14) yields

[S,T ] f (x) =ST f (x)−T S f (x)

=−2πin∑

α,β=1eβeαxα

∂

∂xβf (x)+2πi

n∑β,α=1

eαeβ∂

∂xβ(xα f (x))

=2πi( n∑α=1

xα∂

∂xαf (x)− ∂

∂xα

(xα f (x)

)+2

n∑α<β=1

eαeβ(xα

∂

∂xβf (x)+xβ

∂

∂xαf (x)

))=2πi

(−n f (x)+2

n∑α<β=1

eαeβ(xα

( ∂

∂xβf (x))+xβ(

∂

∂xαf (x)

))).

Hence ∣∣([S,T ] f , f)∣∣2 = ∣∣⟨⟨[S,T ] f ⟩0, f0⟩

∣∣2 = |2πi n|2‖ f0‖4.

We now consider the term‖(S − s) f ‖2.

Note that 2πi x + s is a vector and hence an element of the Clifford group and furthermore2πi x + s = 2πi x + s, whence (2πi x + s)(2πi x + s) = |2πi x + s|2 = (2πi x + s)2.

‖(S − s) f ‖2 = ((S − s)2 f , f

)=

⟨ n∑α=1

n∑β=1

eβeα

∫Rn

(−2πi xα− sα)(−2πi xβ− sβ) f (x) f (x)d x⟩

0

=∫Rn

(−2πi x − s)(−2πi x − s)| f (x)|2d x =∫Rn

|−2πi x − s|2| f (x)|2d x

=∫Rn

n∑α=1

(2πi xα+ sα)2| f (x)|2d x

= 4π2∫Rn

|x + r |2| f (x)|2d x.

Furthermore,

‖(T − t ) f ‖2 = ‖(S − t ) f ‖2 =−∫Rn

n∑α=1

(2πi xα+ tα)2| f (x)|2d x

= 4π2∫Rn

|x + l |2| f (x)|2d x.

Equality holds if

‖⟨(T − t ) f ⟩α‖‖⟨(S − s) f ⟩β‖ = ‖⟨(T − t ) f ⟩β‖‖⟨(S − s) f ⟩α‖,

which holds if f is a radial function. Furthermore,

∃λ ∈R+0 : (S − s) f = iλ(T − t ) f ,


which holds if f (x) = e−πλ

∑nα=1 x2

α , for some λ ∈R+.

Remark 6.3.11 (A shortcut)The same inequality could have been derived by Theorem 6.3.2 as well. Indeed the idea ofthe proof for Theorem 6.3.2 came from a proof of Theorem 6.3.10 which can be found in [17]Corollary 2.8. There Theorem 6.3.10 is proven using Corollary 6.3.1, which yields

4π

√n∑α=1

∫R

(xα+ rα)2| f (x)|2d x

√∫R

(ξα+ lα)2| f (ξ)|2dξ≥ n‖ f ‖2, ∀α= 1, . . . ,n.

Applying the Cauchy-Schwartz inequality yields

4π

√n∑α=1

∫R

(xα+ rα)2| f (x)|2d x

√√√√ n∑β=1

∫R

(ξβ+ lβ)2| f (ξ)|2dξ≥ n‖ f ‖2.

6.3.4 The Riesz transform and Heisenberg uncertainty

We will now compute the effect of the Riesz transform on the appropriate version of the uncer-tainty relation Theorem 6.3.10 derived from Theorem 6.3.2. We cannot study the effects of theRiesz transform directly on Theorem 6.3.10, since the Riesz transform is vector valued.

The essence of the next theorem is that the Heisenberg uncertainty relation is not invariantunder the Riesz transform since the operator S does not commute with Riesz transforms.

Theorem 6.3.12 (The Heisenberg uncertainty relation and the Riesz transform)LET f ∈ L2(Rn ,R).

THEN

(i) 16π2∫Rn

|x + l |2n∑α=1

∣∣Rα f (x)∣∣2d x

∫Rn

|x + r |2n∑α=1

|Rα f (x)|2d x ≥ n2‖ f ‖4;

(ii)∫Rn |x+ l |2 ∑n

α=1 |Rα f (x)|2d x = ∫Rn |x+ l |2| f (x)|2d x, i.e. localisation in the frequency do-

main is invariant under the Riesz transform.

(iii) Let f ,Rα f ∈ W 1,2(Rn) := f ∈ L2(Rn) : ∂α f ∈ L2(Rn),∀α ∈ 1, . . . ,n

. (Sufficient condi-

tions for this to hold will be given in Theorem 5.4.3.)

Then∫Rn |x+r |2 ∑n

α=1 |Rα f (x)|2d x = ∫Rn |x+r |2| f (x)|2d x+ (n−1)

4π2 ‖ 1| · | f ‖2, i.e., localization

in the space domain is not invariant under the Riesz transform.


Proof.

ad(i )n∑

j ,α=1

∣∣⟨[S j ,T j ]Rα f ,Rα f⟩∣∣≥ n

n∑α=1

∥∥Rα f∥∥2 = n

∫Rn

n∑α=1

x2α

|x|2∣∣ f (x)

∣∣2d x

= n‖ f ‖2.

ad(i i )n∑

j ,α=1

∫Rn

(x j + l j )i xα|x| f (x)(x j + l j )

i xα|x| f (x)d x

=n∑

j=1

∫Rn

n∑α=1

x2α

|x|2 (x j + l j ) f (x)(x j + l j ) f (x)d x

=∫Rn

|x + l |2| f (x)|2d x.

ad(i i i )

The additional condition makes sure that S j f ∈ L2(Rn) and S j Rα f ∈ L2(Rn).

n∑j ,α=1

∫Rn

(x j + r j )Rα f (x)(x j + r j )Rα f (x)d x

= 1

4π2

n∑j ,α=1

∫Rn

( ∂

∂x j+2πr j

) i xα|x| f (x)

( ∂

∂x j+2πr j

) i xα|x| f (x)d x

= 1

4π2

n∑j ,α=1

∫Rn

( i xα|x|

( ∂

∂x j+2πr j

)− i x j xα

|x|3 + iδ( j ,α)

|x|)

f (x)

×( i xα|x|

( ∂

∂x j+2πr j

)− i x j xα

|x|3 + iδ( j ,α)

|x|)

f (x)d x

=: I =∫Rn

|x + r |2| f (x)|2d x + (n −1)

4π2

∥∥| · |−1 f∥∥2.

To see this we compute I as

I = 1

4π2

n∑j ,α=1

∫Rn

( i xα|x|

( ∂

∂x j+2πr j

)− i x j xα

|x|3 + iδ( j ,α)

|x|)

f (x)

×( i xα|x|

( ∂

∂x j+2πr j

)− i x j xα

|x|3 + iδ( j ,α)

|x|)

f (x)d x

=∫Rn

|x + r |2| f (x)|2d x + 1

4π2

( n∑j ,α=1

∫Rn

i x j xα|x|3 f (x)

i x j xα|x|3 f (x)d x︸︷︷︸

(1)

+n∑

j ,α=1

∫Rn

iδ( j ,α)

|x| f (x)iδ( j ,α)

|x| f (x)d x︸︷︷︸(2)

+2∑j ,α

∫Rn

i xα|x|

( ∂

∂x j+2πr j

)f (x)

iδ( j ,α)

|x| f (x)d x︸︷︷︸(3)

−2∑j ,α

∫Rn

iδ( j ,α)

|x| f (x)i x j xα|x|3 f (x)d x︸︷︷︸

(4)

−2∑j ,α

∫Rn

i xα|x|

( ∂

∂x j+2πr j

)f (x)

i x j xα|x|3 f (x)d x︸︷︷︸

(5)

).


We continue to compute the terms (1)–(5)

(1) =n∑

j ,α=1

∫Rn

i x j xα|x|3 f (x)

i x j xα|x|3 f (x)d x =∑

j ,α

∫Rn

x2j

|x|2x2α

|x|21

|x|2 | f (x)|2d x =∫Rn

1

|x|2 | f (x)|2d x.

(2) =n∑

j ,α=1

∫Rn

iδ( j ,α)

|x| f (x)iδ( j ,α)

|x| f (x)d x = n∫Rn

1

|x|2 | f (x)|2d x.

(3) = 2∑j ,α

∫Rn

i xα|x|

( ∂

∂x j+2πr j

)f (x)

iδ( j ,α)

|x| f (x)d x = 2∑α

∫Rn

i xα|x|2

( ∂

∂xα+2πr j

)f (x) f (x)d x.

(4) = 2∑j ,α

∫Rn

iδ( j ,α)

|x| f (x)i x j xα|x|3 f (x)d x = 2

∑α

∫Rn

x2α

|x|21

|x|2 | f (x)|2d x = 2∫Rn

1

|x|2 | f (x)|2d x.

(5) = 2∑j ,α

∫Rn

i xα|x|

( ∂

∂x j+2πr j

)f (x)

i x j xα|x|3 f (x)d x = 2

∑j

∫Rn

i x j

|x|2( ∂

∂x j+2πr j

)f (x) f (x)d x.

Whence (1)+(2)-(4)= (n−1)4π2 ‖ 1

| · | f ‖2 and (3)-(5)= 0 and hence

I =∫Rn

|x + r |2| f (x)|2d x + (n −1)

4π2

∥∥∥ 1

| · | f∥∥∥2

.

6.3.5 The affine uncertainty

We now consider the uncertainty relation for the infinitesimal operators of the affine group.

The affine group A is defined in one dimension by

A= (a,b) ∈R2 : a 6= 0

with group law

(a,b) · (c,d) = (ac, ad +b).

As an extension of the affine group of Rn , n ≥ 2 we consider the group of translations andanisotropic dilations. (Note that there are alternative choices for an affine group in higherdimensions. See for example [3].)

The representation of this group on L2(Rn) is given by the operator

Ua,b f (x) =n∏

k=1|ak |

12 f

(x1−b1

a1...

xn−bnan

).


The infinitesimal operators are

A f (x) :=n∑α=1

eα∂

∂aαUa,b f (x)

∣∣∣(a,b)=(1,0)

=n∑α=1

eα( ∏

l=1,...,n|al |− 1

2−aα

2|aα| 32

f(

x1−b1a1...

xn−bnan

)− xα

a2α

n∏l=1

|al |12∂

∂xαf(

x1−b1a1...

xn−bnan

))

=n∑α=1

eα(− 1

2f (x)−xα

∂

∂xαf (x)

)B f (x) :=

n∑α=1

eα∂

∂bαUa,b f (x)

∣∣∣(a,b)=(1,0)

=n∏

k=1|ak |

12

n∑α=1

eα∂

∂xαf(

x1−b1a1...

xn−bnan

)−1

aα

=−n∑α=1

eα∂

∂xαf (x).

The following theorem is the corresponding uncertainty relation.

Theorem 6.3.13 (An affine uncertainty relation)LET f ∈W 1,2(Rn).

THEN

4∥∥∥ n∑α=1

∂α f∥∥∥2∥∥∥n

2f +

n∑α=1

· α∂α f∥∥∥2 ≥

∣∣∣⟨1

2

n∑α=1

∂α f , f⟩∣∣∣2

, (6.21)

where ∂α f (x) := ∂∂xα

f (x), ∀x ∈Rn .

Proof. We need to compute [A,B ], so we start by considering

ABe0 f (x) =n∑

α,β=1eβeα

∂

∂xα(−1

2−xβ

∂

∂xβ) f (x)

=n∑α=1

∂

∂xα(

1

2+ ∂

∂xαxα+ 1

2)+

n∑α,β=1

eαeβ∂

∂xα(

1

2+xβ

∂

∂xβ) f (x)

and continue with

B Ae0 f (x) =n∑

α,β=1eαeβ(−1

2−xα

∂

∂xα)∂

∂xαf (x)

=n∑α=1

(1

2+xα

∂

∂xα)∂

∂xαf (x)−

n∑α,β=1

eαeβ∂

∂xα(

1

2+xβ

∂

∂xβ) f (x).

Consequently

[A,B ]e0 f (x) = (AB −B A) f (x) = 1

2

∂

∂xαf (x)+2

n∑α,β=1

eαeβ∂

∂xα(

1

2+xβ

∂

∂xβ) f (x).

6.4. DISCUSSION AND COMPARISON TO THE LITERATURE 129

Hence ([A,B ]e0 f ,e0 f ) = ⟨ 12

∑nα=1

∂∂ · α f , f ⟩. Applying (6.6) yields

4∥∥∥ n∑α=1

∂

∂ · α∥∥∥2∥∥∥n

2+

n∑α=1

· α∂

∂ · α∥∥∥2 ≤

∣∣∣⟨1

2

n∑α=1

∂

∂ · αf , f ⟩

∣∣∣2. (6.22)

6.4 Discussion and comparison to the literature

The first new result in this chapter is Theorem 6.2.2. It states that under the assumption of onevanishing moment the Heisenberg uncertainty relation is invariant under the Hilbert trans-form. Of course the question arises if a the same holds for the Riesz transform. Since the Riesztransform R is unitary (see Theorem 4.1.4.) whereas the partial Riesz transforms Rα are not itseems reasonable to consider a single uncertainty relation for the Riesz transform rather thana set of uncertainty relations for the partial Riesz transforms. This requires a new kind of un-certainty relation for vector valued functions.

We consider two approaches to such uncertainty relations for vector valued functions - a classi-cal approach Theorem 6.3.2 based on the Cauchy-Schwartz inequality and a second approachon Hilbert-Clifford modules Theorem 6.3.3. The second approach is not translation invariantthus we give three special cases which are translation invariant - Corollary 6.3.6, Corollary 6.3.8and Corollary 6.3.9. Theorem 6.3.2 is easier to use and gives stronger inequalities than Theo-rem 6.3.3 and its special cases. However Theorem 6.3.3 is more general.

Theorem 6.3.10 is an example of a single uncertainty relation for operators in higher dimen-sions known in the literature (see [17] Corollary 2.8.) - however we present the first systematicapproach. In [11] an uncertainty relation for vector-valued operators is discussed however ityields different uncertainty relations than the ones we consider and does not result in a singleuncertainty relation in higher dimensions.

In Theorem 6.3.12 we apply Theorem 6.3.2 to the Riesz transform of a function and get aHeisenberg uncertainty relation for the Riesz transformed function. Furthermore we showthat while the frequency localization is invariant under the Riesz transform the localization inspace domain is not and we give a formula for the localization in space domain.

As a further application of Theorem 6.3.3 we give an affine uncertainty relation Theorem 6.3.13that is based on the tensor approach to the affine group and is to our knowledge entirely new.

Chapter 7

The Gibbs phenomenon for waveletframes

7.1 Preliminaries

We like to investigate the existence of Gibbs’ phenomenon for wavelet frames which are basedon some kind of multiresolution analysis, i.e., for which a scaling function exists that satisfiescondition (7.1).

Figure 7.1: The Gibbs phenomenon demonstrated by partial sums Sn of the Fourier seriesfor the characteristic function of the interval [−π/2,π/2]: S1 =green, S5 =orange, S10 =red,S40 =magenta, S100 =blue, S500 =black

The Gibbs phenomenon was first described for partial sums of Fourier series. (See [27] forfurther details.) It demonstrates the difficulties in approximating functions with jump dis-continuities using partial Fourier series. The Gibbs phenomenon describes the appearance ofovershoots and undershoots near jump discontinuities as shown in Figure 7.1.

131

132 CHAPTER 7. THE GIBBS PHENOMENON

Here the characteristic function of the interval [−π/2,π/2] is approximated by the partial Fourierseries

Sn(x) :=n∑

k=0

∫[−π/2,π/2]

e−2πi kt d te2πi kx , ∀x ∈ [−π,π]

This phenomenon is not restricted to Fourier series but also occurs in other representations offunctions containing jump discontinuities such as the representations using general orthogo-nal series expansions, general integral transforms, spline approximations, and continuous aswell as discrete wavelet approximations. (See [27] for further details.)

In [44] the Gibbs phenomenon for orthonormal wavelet bases based on a MRA is studied. Ourapproach is to extend certain results of [44] from orthonormal wavelet bases to a certain classof wavelet frames.

The main contribution here is the observation that condition (7.1) is needed for the proof thatthe Gibbs phenomenon occurs. It is automatically satisfied by wavelet orthonormal bases.

In order to state a formal definition of the Gibbs phenomenon we will need some notation. LetA ∈ GL(n) and let

ψ j ,k j∈Z,k∈Zn := A j Tkψ j∈Z,k∈Zn ⊂ L2(Rn)

be a wavelet frame with dual frame

ψdj ,k j∈Z,k∈Zn := A j Tkψ

d j∈Z,k∈Zn ⊂ L2(Rn)

such that there exist scaling functions φ,φd ∈ L2(Rn) which satisfy∑j∈N

∑k∈Zn

⟨ f , A− j Tkψ⟩A− j Tkψd = ∑

k∈Zn

⟨ f ,Tkφ⟩Tkφd . (7.1)

Remark 7.1.1 (Sufficient conditions for (7.1))LET φ ∈ L2(Rn) be a refinable function with refinement mask H0 such that φ is continuousat the origin and lim|ξ|→0 φ(ξ) = 1. Furthermore for let H1 ∈ L∞(T) be the wavelet mask forψ ∈ L2(Rn), where ψ is the mother wavelet of a tight wavelet frame with frame bound A = 1.

Moreover, suppose that for all q ∈D−1a Z

n/Zn and for almost all ξ,ξ+q ∈Twith∑k∈Zn

|φ(ξ+k)|2 > 0 ∧ ∑k∈Zn

|φ(ξ+q +k)|2 > 0

equation (5.9) hold:H0(ξ)H0(ξ+q)+H1(ξ)H1(ξ+q) = δq,0.

THEN Theorem 5.5.14 states that condition (7.1) is satisfied, where φ=φd and ψ=ψd .

Proposition 7.1.2LET φ,φd ,ψ,ψd be such that condition (7.1) holds. Let

q(x, t ) := ∑k∈Zn

φ(x −k)φd (t −k)

and qm(x, t ) := |det (A)|m q(Am x, Am t ).

THEN

fm(x) : =∫Rn

qm(x, t ) f (t )d t =∫Rn

∑k∈Zn

φ(Am x −k)φd (Am t −k) f (t )d t

= ∑k∈Zn

⟨ f , AmTkφ⟩AmTkφd (x)

(7.1)= ∑k∈Zn , j<m

⟨ f , A j Tkψ⟩A j Tkψd (x)

and hence fmm→∞−→ f .

7.2. THE GIBBS PHENOMENON 133

7.2 The Gibbs phenomenon

EXAMPLE 7.2.1 (Gibbs phenomenon in one dimension):LET f ∈ L2(R) be continuous in a neighborhood of some a ∈R except for a jump at a. Let

a− := limt→a,t<a

f (t ) and a+ := limt→a,t>a

f (t ).

Without loss of generality let a− < a+.

THEN the dual frames generated by ψ,ψd ∈ L2(R) are said to exhibit a Gibbs phenomenon ata if either ∃al l∈N : al < a and limm→∞ fm(am) < a− or ∃al l∈N : al > a limm→∞ fm(a) > a+.

In the case that the dimension is n > 1 we will use the following definition of the Gibbs phe-nomenon.

Definition 7.2.1 (Gibbs phenomenon)Let f :Rn →Rhave the property that there exists a (n−1)–dimensional hypersurface S satisfying0 ∈ S and that f is continuous ∀x ∈Rn \ S. Let this hypersurface S be given by a function g ∈C 1(Rn) with the property that ∇g (0) 6= 0 via g (x) = 0 ⇔ x ∈ S. It follows, that g (0) = 0 and f iscontinuous ∀x ∈Rn such that g (x) 6= 0.

There exists a radius r > 0 such that the surface S divides the ball Br (0) into two connected partsB+ := x ∈ Br (0) : g (x) > 0 and B− := x ∈ Br (0) : g (x) ≤ 0.

f is called piecewise continuous on Br (0) if there exist two continuous functions f +, f − onBr (0) such that f (x) = f ±(x), ∀x ∈ B±.

f is said to have a jump discontinuity at 0 if f +(0) 6= f −(0).

LET f : Rn → R be a be a bounded, measurable function which is piecewise continuous andexhibits a jump discontinuity at 0.

Let ∇g (x) := ( ∂∂x1

g (x), . . . , ∂∂xn

g (x)) and

γ := ∇g (0)

|∇g (0)| . (7.2)

Let µ be a unit vector such that ⟨γ,µ⟩ 6= 0.

THEN the wavelet approximation fm is said to show a Gibbs phenomenon at 0 in the direc-tion of µ, if there exists a sequence amm∈N ⊂R+ such that am → 0 as m →∞ and

either L > limm→∞ f (amµ), when lim

m→∞ f (amµ) > limm→∞ f (−amµ)

or L < limm→∞ f (amµ), when lim

m→∞ f (amµ) < limm→∞ f (−amµ),

where L = l i mm→∞ fm(amµ).

Definition 7.2.2 (l -regular functions)A function f ∈C l (Rn) is called l-regular in the sense of Mallat, or simply l-regular, if

∀α= (α1, . . . ,αn) ∈Nn0 , |α| ≤ l , ∀k ∈N∃Ck > 0 : |∂α f (x)| ≤Ck (1+|x|)−k , ∀x ∈Rn .

Henceforth we will assume that φ,φd are l -regular for some l ∈N0.

It follows that

∀|α|, |β| ≤ l , ∀k ∈N∃Ck > 0 : |∂αx ∂βt q(x, t )| ≤Ck (1+|x − t |)−k , ∀x, t ∈Rn . (7.3)


To show that the formulation of the Gibbs phenomenon is meaningful we show that the Gibbsphenomenon does not occur at points, where the function f is continuous. This theoremextends a similar theorem for wavelet orthonormal bases given in [44].

Theorem 7.2.3LET f :Rn →R be bounded and measurable. Let K ⊂Rn be compact and let f be continuousfor every x ∈ K .

THEN fm(x)m→∞−→ f (x), uniformly ∀x ∈ K .

Proof. By the regularity of q it is clear, that fm(x) converges uniformly, whence limm→∞ fm isa continuous function. Indeed let C := supx∈Rn

∫Rn |q(x, t )|d t . Since K is compact and f is

continuous on K ,

∀ε> 0∃δ> 0 : | f (t )− f (x)| < ε

2C,∀x ∈ K , t ∈Rn : |x − t | < δ.

For x ∈ K

| fm(x)− f (x)| =∣∣∣∫Rn

qm(x, t )( f (t )− f (x))d t∣∣∣

≤∫|t−x|<δ

|qm(x, t )| | f (t )− f (x)|d t +∫|t−x|≥δ

|qm(x, t )| | f (t )− f (x)|d t

= I + J

Now we estimate I as

I =∫|t−x|<δ

|qm(x, t )|| f (t )− f (x)|d t ≤ ε

2C

∫|t−x|<δ

|det(Am)||q(Am x, Am t )|d t

= ε

2C

∫|A−m t−x|<δ

|q(Am x, t )|d t

≤ ε

2.

Choosing k > n in (7.3), we continue by estimating J by

J =∫|t−x|≥δ

|qm(x, t )| | f (t )− f (x)|d t ≤∫|t−x|≥δ

|det(Am)| |q(Am x, Am t )|2‖ f ‖∞d t

≤ |det(Am)|∫|t−x|≥δ

Ck

(1+|Am(t −x)|)k2‖ f ‖∞d t

=∫|A−m t |≥δ

Ck

(1+|t |)k2‖ f ‖∞d t

m→0−→ 0.

It follows that for any ε > 0 there exists some m0 ∈N such that | fm(x)− f (x)| < ε, ∀m ≥ m0,x ∈ K .

7.3. SUFFICIENT CONDITIONS 135

7.3 Sufficient conditions for the occurrence of the Gibbs phe-nomenon

The following theorem states sufficient conditions for a Gibbs phenomenon to occur for a cer-tain very simple model function.

Theorem 7.3.1 (Sufficient conditions)LET φ,φd :Rn →R be refinable functions such that (7.1) is fullfilled. Let µ, and γ, be eigen-vectors of A, and A∗, respectively, with the same positive eigenvalue, which fullfill |µ| = |γ| = 1

and ⟨γ,µ⟩ > 0. Finally let f (x) = H(⟨γ, x⟩), where H(x) :=

1 x ≥ 0

0 x < 0is the Heaviside function.

THEN the wavelet approximation fm exhibits a Gibbs phenomenon at 0, iff∫⟨γ,t⟩>0

q(aµ, t )d t > 1 for some a > 0. (7.4)

Proof. The proof is the same as in the orthonormal bases case found in [44].

fm(x) =∫⟨γ,t⟩>0

|det(Am)|q(Am x, Am t )d t =∫⟨γ,A−m v⟩>0

q(Am x, v)d v

We know that ∃λ> 0 : A∗γ=λγ and Aµ=λµ. Since

⟨γ, v⟩ =⟨

(A∗)mγ, A−m v⟩=

⟨λmγ, A−m v

⟩,

we have that for every b ∈R

fm(bµ) =∫⟨γ,v⟩>0

q(bλmµ, v)d v. (7.5)

Let us assume that (7.4) holds. Set b = am = aλ−m to obtain

fm(amµ) =∫⟨γ,v⟩>0

q(aµ, v)d v > 1.

Hence fm(amµ) is a constant sequence whith value greater than 1. Since

1 = limt→0, t>0

f (t ) > limt→0, t<0

f (t ) = 0

we have shown a Gibbs phenomenon in direction µ to exist.

Conversely assume, that

∃amm∈N : 0 > amm→∞−→ 0, such that lim

m→∞ fm(amµ) > 1.

Then ∃m ∈N, fm(amµ) > 1. Setting b = amλ−m in (7.5) we get (7.4).

Lemma 7.3.2LET f :Rn →R be bounded and measurable, and let A be a dilation matrix all of whose eigen-values have the same absolute value. Consider the cone

C := x ∈Rn : |⟨x,µ⟩| ≥ |x|cos(θ)

,

where θ ∈]0, π2 ] and µ ∈Rn , |µ| = 1. Furthermore, let f (x) = 0, ∀x ∈C , and limb→0,b>0 f (by) =0, for almost all y ∈Rn .

THEN fm(aµ)m→∞−→ 0 uniformly for a ∈R.


Proof.

fm(x) =∫Rn

qm(x, t ) f (t )d t =∫Rn \C

qm(x, t ) f (t )d t .

Let x = aµ with a ∈R. Let t ∈Rn \C and k > n.

Then

|x − t | =√

a2 +|t |2 −2a⟨µ, t⟩ ≥√

a2 +|t |2 −2|a| |⟨µ, t⟩|≥

√a2 + t 2 −2a|t |cos(θ) ≥

√a2 + t 2 −a2 − t 2 cos2(θ)

= |t |sin(θ)

and consequently

|qm(x, t )| (7.3)≤ Ck∣∣det(Am)

∣∣(1+|Am(x − t )|)−k ≤Ck∣∣det(Am)

∣∣(1+ ∣∣det(Am)∣∣|x − t |)−k

≤Ck |det(Am)|(1+ ∣∣det(Am)∣∣ |t |sin(θ)

)−k .

Applying this to fm we get

| fm(aµ)| ≤Ck

∫Rn

∣∣det(Am)∣∣(1+ ∣∣det(Am)

∣∣|t |sin(θ))−k f (t )d t

=Ck

∫Rn

(1+|t |sin(θ))−k f(∣∣det(Am)

∣∣−1t)d t .

By the Lebesgue dominated convergence theorem the last expression converges uniformly tozero as m →∞.

Theorem 7.3.3LET f :Rn →R be a bounded measurable function that is piecewise continuous at 0. Let γ asin (7.2) and let µ ∈Rn : |µ| = 1, ⟨µ,γ⟩ > 0.

THEN limm→∞ fm exhibits a Gibbs phenomenon at 0 iff (7.4) holds.

Proof. Let r,B± and f ± be as in Definition 7.2.1. Replacing f by a linear combinationα f (x)+βif necessary, we may assume that f +(0) = 1 and f −(0) = 0. Let θ ∈]0, π2 [: θ < π

2 −ψ, where ψ isthe angle between γ and µ. Let

C± := x ∈Rn :

∣∣⟨x,µ⟩∣∣≥ cos(θ)|x| and ±⟨x,µ⟩ > 0.

C± lies on opposite sides of the tangent plane γ⊥ = x ∈Rn : ⟨x,γ⟩ = 0

. Then C±∩Br (0) ⊂ B±.

Finally let F :Rn →R, x 7→

1, ∀x ∈ B+∪C+;

0, otherwise.

Then there exists an ε> 0 such that f −F is continuous for all aµ such that a < ε. Therefore, byTheorem 7.2.3 and by the linearity of the wavelet transform, fm shows a Gibbs phenomenonin the direction of µ iff Fm does. Now consider G(x) := F (x)−H(γ · x). Then G satisfies all hy-

potheses for Lemma 7.3.2, hence Gm(aµ)m→0−→ 0 uniformly for a ∈R. Consequently fm exhibits

a Gibbs phenomenon iff Hm(γ · · ) does. Now Theorem 7.3.1 gives the result.

Chapter 8

Higher Riesz transforms andrepresenations of the rotation group

A wavelet system consists of a set of functions derived from a mother wavelet by theapplicationof a set of operators which are a unitary representation of a certain group - in the one dimen-sional case this is the affine group on L2(R). These sets are then sampled to give a basis or aframe for L2(Rn). Riesz transforms are very well suited for an implementation via wavelets. Inthis chapter we show that this is due to the fact that the steering corresponds to a unitary rep-resentation of the rotation group on the set of Riesz transformed functions. For n > 1 there arean infinite number of such unitary representations of SO(n) based on the spherical harmonicswhich yield steerable operators - the higher Riesz transforms. We give explicit constructions ofhigher Riesz transforms for the interesting cases n = 2,3.

New contributions in this chapter are first of all the definition of higher Riesz transforms as thevector of partial higher Riesz transforms Definition 8.2.4 corresponding to a basis of sphericalharmonics Hk . These higher Riesz transforms are the smallest possible sets of partial higherRiesz transforms which are invertible and steerable. Combining these minimal higher Riesztransforms in Corollary 8.2.18 we derive higher Riesz transforms which are geometrically richerat the cost of higher redundancy. In Remark 8.2.19 we learn how to construct Higher Riesztransforms which contain the directional information of higher derivatives. This approach issimilar to the steerable pyramids of Freeman and Adelson in [18] – indeed in Remark 8.2.22 weshow that these steerable pyramids have a close connection to higher Riesz transforms.

To proof that the higher Riesz transforms map frames onto frames in Theorem 8.2.7 we firstshow that the hypercomplex higher Riesz transforms defined in Definition 8.2.4 are unitaryoperators. Using our result Theorem 5.2.13, we conclude in Theorem 8.2.8 that the higherRiesz transforms map Clifford frames onto Clifford frames.

A key property for image analysis of the Riesz transform is the decomposition of an image intoamplitude, phase and phase direction. In Definition 8.2.9 we state the concept of phase direc-tion for higher Riesz transforms. Based on the concept of hypercomplex higher Riesz trans-forms we define a higher monogenic signal in Definition 8.2.11 which yields an amplitude-phase decomposition for higher Riesz transforms.

Theorem 8.2.15 yields the connection of the higher monogenic signal to certain generalizedCauchy Riemann equations derived from higher Dirac operators Definition 8.2.13 which arethe square-root of a power of the Laplace operator.

137

138 CHAPTER 8. HIGHER RIESZ TRANSFORMS

8.1 Definitions and basic theorems

We will start with some basic definitions and theorems which can be found in similar form in[33, 45, 14] and [57].

8.1.1 Definition of spherical harmonics

Definition 8.1.1 (Linear spaces of homogenous polynomials)A function f on a vector space V is called homogenous of degree k ∈N iff

∀ε> 0 f (εx) = εk f (x), ∀x ∈V.

Let V =Rn . The space of homogenous polynomials of order k is denoted by

Pk (Rn) :=

p(x) = ∑|α|=k

aαxα, p is homogenous of degree k

,

where α ∈Nn0 is a multiindex and n is the dimension of the vector space V .

Let f ∈C 2(V ). The Laplace differential operator is given by

4 f (x) =n∑

l=1

d 2

d x2k

f (x), ∀x ∈V.

The solid spherical harmonics are the subspace of the homogenous polynomials of degree kthat lies in the kernel of 4:

Hk (Rn) = p ∈Pk : 4p = 0

.

The surface spherical harmonics are the restriction of the solid spherical harmonics to theunit sphere:

Hk := p|Sn−1 , p ∈Hk

.

Remark 8.1.2For the rest of this chapter k ∈N0 will denote the order of the polynomial spaces considered.

Of course Hk is a linear subspace of L2(Sn−1).

A scalar product on L2(Sn−1) is given by

⟨P,Q⟩ :=∫

Sn−1P (x)Q(x)ω−1

n dσ(x), ∀P,Q ∈ L2(Sn−1),

where ωn = 2πn/2

Γ(n/2) is the surface area of the unit sphere.

8.1.2 Properties of spherical harmonics

Theorem 8.1.3 (Orthogonality of spherical harmonics)The finite dimensional spaces Hk , k ∈N0 are mutually orthogonal with respect to the scalarproduct on L2(Sn−1).

Proof. This is 3.1.1 in [45].

8.1. DEFINITIONS AND BASIC THEOREMS 139

The following theorem gives a relation between homogenous polynomials and spherical har-monics.

Theorem 8.1.4 (Dimension of spherical harmonics)LET p ∈Pk .

THEN there exist p1 ∈Hk , p2 ∈Pk−2 such that p(x) = p1(x)+|x|2p2(x), ∀x ∈Rn . The dimen-sion of Pk is

dim(Pk ) =(

n +k −1

k

),

whence the dimension of the spaces of spherical harmonics is

dn,k := dim(Hk (Rn)) = dim(Hk (Rn)) =(

n +k −1

k

)−

(n +k −3

k −2

).


Theorem 8.1.5 (Completeness of spherical harmonics)The collection of all finite linear combinations of elements of Hk (Rn) is

(i) dense in(C (Sn−1),‖ · ‖∞

);

(ii) dense in L2(Sn−1).

Proof. This is IV.2.3 in [46].

Definition 8.1.6 (Some orthogonal polynomials)The Gegenbauer polynomials also called ultraspherical polynomials Cν

k of degree k andorder ν 6= 0 are generated by the function

(1−2xt + t 2)−ν =∞∑

k=0Cν

k (x)t k , ∀x, t ∈R.

The Legendre polynomial of degree k is defined by

Pk (x) = 2−k

k !

d k

d xk(x2 −1)k , ∀x ∈R.

The Tchebichef polynomial of degree k is generated by

−1/2log(1−2xt + t 2) = ∑k∈N0

(k +1)−1Tk+1(x)t k+1, ∀x, t ∈R.

Theorem 8.1.7 (Explicit formulas for orthogonal polynomials)

(i) LET 2ν ∈N and let l = bνc ∈N0.

THEN

Cνk (x) =C l+1/2

k (x) = 2l l !

(2l )!

d l

d xlPn+l (x) = 2−k l !

(2l )!(k + l )!

d k+2l

d x+2l(x2 −1)k+l , ∀x ∈R,


respectively,

Cνk (x) =C l+1

k (x) = 2−l

l !(k + l +1)

d l+1

d xl+1Tk+l+1(x), ∀x ∈R.

(ii) LET l ∈N0.

THEN

C l/2k (1) = (k + l −1)!

k !(l −1)!=

(k + l −1

k

).

(iii) The Tchebichef polynomials are given as

Tk (x) = 1

2

((x + i (1−x2)

12)k + (

x − i (1−x2)12)k

)= cos

(k cos−1(x)

), ∀x ∈R.

Theorem 8.1.8 (Addition theorem)

LET y lk

dn,k

l=1 ⊂ Hk be a real orthonormal basis of Hk with respect to the scalar product on

L2(Sn−1).

THEN for any fixed η ∈ Sn−1

Cn−2

2k (⟨ξ,η⟩)C

n−22

k (1)= 1

dn,k

dn,k∑l=1

y lk (ξ)y l

k (η), ∀ξ ∈ Sn−1.

Especiallydn,k∑l=1

|y lk (ξ)|2 = dn,k . (8.1)

Proof. See [14], Chapter 11.4.

Lemma 8.1.9 (Representation of the rotation group)

LET

y lk

dn,k

l=1 ⊂Hk be an orthonormal basis of Hk and let ρ ∈ SO(n).

THEN ∃Dρ,k = (Dl ,r )dn,k

l ,r=1 ∈ SO(dn,k ) such that

y lk (ρξ) =

dn,k∑r=1

Dl ,r y rk (ξ).

Dρ,k is a representation of the rotation group.

Proof. See [14], Chapter 11 Lemma 5.

8.1.3 The spaces Hk (Rn)

Definition 8.1.10 (The spaces Hk (Rn))Hk (Rn) :=

f ∈ L2(Rn) : f (x) = g (|x|)hk (x), where hk ∈Hk (Rn), g :R+ →R.

8.2. HIGHER RIESZ TRANSFORMS 141

Theorem 8.1.11 (Completeness of the spaces Hk )It holds that L2(Rn) =⊕

k∈N0 Hk (Rn) in the following sense:

(i) Lethk

1 , . . . ,hkdn,k

be an orthonormal basis of Hk .

Then any f ∈ Hk (Rn) can be written as

f (x) =dn,k∑j=1

f j (|x|)hkj (x).

It follows that ∫Rn

| f (x)|2d x =dn,k∑j=1

∫R+

0

| f j (r )|2r 2k+n−1dr.

(ii) Each subspace Hk (Rn) is closed.

(iii) The Hk (Rn) are mutually orthogonal.

(iv) Every element of L2(Rn) is a limit of finite linear combinations of elements belonging tothe spaces Hk (Rn).


Corollary 8.1.12 (Invariance under the Fourier transform)The spaces Hk are invariant under the Fourier transform, i.e., F : Hk (Rn) → Hk (Rn).

LET p ∈Hk and f :R→R such that p f (| · |) ∈ L2(Rn).

THEN there exists a function g :R→R such that

F (p f )(ξ) = p(ξ)g (|ξ|), f.a.a. ξ ∈Rn .

Proof. This is a corollary of Theorem 3.4 in [45].

8.2 Higher Riesz transforms and irreducible representationsof the rotation group

Remark 8.2.1 (An algebra of Riesz transforms)Since the partial Riesz transforms Rα, α ∈ 1, . . . ,n, map Lp (Rn) to Lp (Rn) for1 < p < ∞, they span an algebra A of linear bounded operators, invariant under translationand dilation. Let α,β ∈Nn

0 , a ∈R. The algebra is generated by

· :A×A→A, F (Rα ·Rβ f )(ξ) = i |α|ξαξβ

|ξ||α|+|β| f (ξ),

+ :A×A→A, F((Rα+Rβ) f

)(ξ) =

( i |α|ξα

|ξ||α| + i |β|ξβ

|ξ||β|)

f (ξ),

and multiplication with a scalar is given by

· :R×A→A, aRα f (ξ) = ai |α|ξα

|ξ|α|| f (ξ), ∀ f ∈ Lp (Rn), f.a.a. ξ ∈Rn .


It is evident that the elements of the algebra A are invariant with respect to dilation and trans-lation. Examples of elements of this algebra are the operators defined in Theorem 8.2.2. In thissection we find rotation invariant linear subspaces of A on which there exists a representationof the rotation group.

Theorem 8.2.2 (Fourier multipliers of singular integral operators)LET P ∈Hk (Rn), k ∈N, 1 < p <∞. Furthermore, let the singular integral operator

RP : Lp (Rn) → Lp (Rn)

be given by

RP f (x) := limε→0

∫|y |≥0

γkP (y)

|y |k+nf (x − y)d y, ∀ f ∈ Lp (Rn), f.a.a. x ∈Rn ,

where γk := Γ(

k+n2

)πn/2Γ

(k2

) .

THEN the Fourier multiplier corresponding to RP is

mp (ξ) := i k P (ξ)

|ξ|k , ∀ξ ∈Rn .

That is F(RP f

)= mp f .

Proof. This is theorem 3.5 in [45].

Remark 8.2.3 (Riesz transforms depend on surface spherical harmonics)Every solid spherical harmonic p ∈ Hk may be written at a pointx ∈ Rn as p(x) = |x|k p

( x|x|

). The Fourier multiplier of the operator Rp from Theorem 8.2.2

is p(x)|x|k = p

( x|x|

)– the restriction to the unit sphere of the solid spherical harmonic p and as

such it is a surface spherical harmonic. Hence the operator Rp from Theorem 8.2.2 dependsonly on the surface spherical harmonic p|Sn−1 .

For the rest of the section we will assume thathk

1 , . . . ,hkdn,k

is an ONB for Hk (Rn) in the sense

that ⟨hkl |Sn−1 ,hk

r |Sn−1⟩L2(Sn−1)

= δr,k , ∀l ,r = 1, . . . ,dn,k . It follows from Theorem 8.1.8 that

dn,k∑l=1

(hk

l (x))2 =

dn,k∑l=1

(hk

l

( x

|x|)|x|k)2 = dn,k |x|2k , ∀x ∈Rn .

Definition 8.2.4 (Higher Riesz transform)LET P ∈Hk (Rn), k ∈N.

THEN the singular integral operator RP defined in Theorem 8.2.2 is called partial higher Riesz-transform. k is called the order of the higher Riesz transform.

R = Rk = d−1/2n,k

(Rhl ,k

)dn,k

l=1 is called the higher Riesz transform of order k. The hypercom-plex higher Riesz transform is the corresponding operator on the Clifford-Hilbert moduleL2(Rn)dn,k

given by

Rk : L2(Rn)dn,k→ L2(Rn)dn,k

, f 7→Rk f = d−1/2n,k

dn,k∑l=1

el Rhkl

f .

For notational convenience we will sometimes skip the index of Rk and simply write R instead.


Theorem 8.2.5LET A be the algebra of operators on L2(Rn) algebraically generated by the partial Riesz trans-forms R1, . . . ,Rn .

THEN

(i) every partial higher Riesz transform belongs to A;

(ii) the closure of A in the strong operator topology is identical with the algebra of boundedtransformations on L2(Rn) which commute with translations and dilations.


8.2.1 Higher Riesz transforms and representations of the rotation group

Theorem 8.2.6 (Spherical harmonics and representations of the rotation group)LET V be a finite dimensional Hilbert space and let ρ→ Tρ be a continuous homomorphismfrom SO(n) to the group of unitary transformations on V . Recall from Definition 1.3.28 on page14 that the couple (Tρ ,V ) is called a unitary representation of SO(n).

A unitary representation is irreducible if there is no non-trivial invariant subspace of V underTρ , ρ ∈ SO(n).

Two representations (Tρ ,VT ) and (Sρ ,VS ) are equivalent, iff there is a unitary correspondenceU : VT ↔VS , such that U−1SρU =Tρ .

(i) Let V =Hk . Then the following representation is irreducible:

(TρP (x)) = P (ρ−1x), ρ ∈ SO(n), P ∈Hk .

(ii) An irreducible representation (Sρ ,V ) of SO(n) is equivalent to the one above iff

∃v ∈V : Sρ(v) = v, ∀ρ ∈ SO(n −1).

(iii) Let T : L2(Rn) → L2(Rn ,V ) be bounded and linear.

(a) If T commutes with translations and dilations and if

ρTρ−1( f ) =TρT f ,

then either T = 0 or (Tρ ,V ) is equivalent to the representation given in (i).

(b) If (Tρ ,V ) arises from spherical harmonics of degree k ≥ 1 then T is determined upto a constant multiple. Let β1, . . . ,βN be a basis for the linear functionals on V ,then each β j (T f ) is a partial higher Riesz transform of degree k.

(iv) Let hkl

dn,k

l=1 ⊂Hk be a orthonormal basis of Hk . Hk is isomorphic toRdn,k via hkl 7→ el and

Tρ is isomorphic to a matrix Dρ ∈ SO(dn,k ) - the Wigner D-matrix.

Hence the Wigner D-matrices yield a representation of the rotation group(Dρ ,Rdn,k

).

Proof. (i) and (ii) are III.4.7 in [45]. (iii) is III.4.8 in [45].


EXAMPLE 8.2.1 (Higher Riesz transforms):The spherical harmonics of degree k yield a number of spaces that have dimension dn,k . Inthese spaces exist unitary representations of the rotation group given by unitary automor-phisms that are equivalent to the Wigner D-matrices:

1. Let U :Hk (Rn) →Rdn,k , p =∑dn,kα=1 aαhk

α 7→ (a1, . . . , adn,k).

Then a irreducible representation (Dρ ,Rdn,k ) of the group SO(n) is given by

Sρ =U−1TρU ,

where (Tρ ,Hk ) is the representation given in 8.2.6(i). The matrices Dρ ∈ SO(dn,k ) wereintroduced in [57] by Wigner in and are called Wigner-D matrices.

This representation induces a representation(Sρ ,V

):

2. Let R : L2(Rn) → L2(Rn ,Rdn,k ) be the higher Riesz transform given by

R f = Rhk1

f , . . . ,Rhkdn,k

f .

Let ek := δk,l dn,k

l=1 . Then eT1 , . . . ,eT

dn,k is a basis for Rdn,k∗ and eT

α(R f ( · )) = Rhkα

f is a

partial Higher Riesz transform as is stated in Theorem 8.2.6 iii b).

Let ρ ∈ SO(n), f ∈ L2(Rn). Then

ρRρ−1 f = ρR f (ρ−1 · ) = ρ(Rhk

1, . . . ,Rhk

dn,k

)f (ρ−1 · )

=F−1(ρ( ihk

1

‖ · ‖n , . . . ,ihk

dn,k

‖ · ‖n

)f (ρ−1)

)=F−1

(( ihk1 (ρ · )

‖ · ‖n , . . . ,ihk

dn,k(ρ · )

‖ · ‖n

)f)

=F−1(( iTρh

k1

‖ · ‖n , . . . ,iTρh

kdn,k

‖ · ‖n

)f)= (

RTρhk1

, . . . ,RTρhkdn,k

)f

= SρR f

That is(Sρ ,R(L2(Rn))

)is equivalent to the representation (Tρ ,Hk (Rn)).

Since Sρ is a linear combination of the elements of the vector R f , it can be considered

as a mapping in V =Rdn,k . As such Sρ =Dkρ .

3. Let p =∑dn,k

l=1 alhkl ∈Hk . Note that Rp |H0 : H0(Rn) → Hk (Rn).

Let f ∈ Hk (Rn). Then there exist f1, . . . , fdn,k∈ H0(Rn) and a = (a1, . . . , adn,k

) ∈Rdn,k suchthat

f =dn,k∑m=1

amRhkm

fm .

Hence a representation of the rotation group is given by

Sρ : Hk → Hk , f 7→ Sρ( f ) :=dn,k∑m=1

(Dkρ a)mRhk

mfm .

The representation(Sρ , Hk (Rn)

)is equivalent to the representation (Tρ ,Hk (Rn)).


8.2.2 Higher Riesz transforms of frames

Using the proper normalization, the higher Riesz transform of degree k is a unitary operatoron the Clifford-Hilbert module L2(Rn)dn,k

. As a consequence the higher Riesz transform of the

mother wavelet of a wavelet frame for L2(Rn) is the set of mother wavelets for a multiwaveletframe of L2(Rn). This multiwavelet frame is steerable with respect to rotation iff the originalwavelet frame was derived from a radial, i.e. rotation invariant mother wavelet.

To proof this we will first proof that the hypercomplex higher Riesz transforms are unitary op-erators.

Theorem 8.2.7 (Unitarity of hypercomplex higher Riesz transforms)LET hk

1 , . . . ,hkdn,k

⊂ Hk be a basis of Hk , such thathk

l

∣∣sn−1

l are orthogonal with respect to

⟨ · , · ⟩L2(Sn−1) and normalized such that for all ω ∈ Sn−1

dn,k∑l=1

(hk

l (x))2 = dn,k |x|2k , ∀x ∈Rn . (8.2)

Furthermore, let

R : L2(Rn)dn,k→ L2(Rn)dn,k

, f 7→ d−1/2n,k

∑β∈Odn,k

dn,k∑α=1

eβeαRhkα

fβ.

THEN for k ∈ 2N−1 the operator R is unitary and self adjoint in the Clifford-Hilbert moduleL2(Rn)dn,k

.

For k ∈ 2N the operator iR is self adjoint and R is unitary in the Clifford-Hilbert moduleL2(Rn)dn,k

.

Proof. (8.1) in Theorem 8.1.8 on page 140 states that we can indeed normalize any basis ofspherical harmonics whose elements are orthogonal to each other such that (8.2) is fulfilled.

Since the polynomials hkm are k-homogenous, it follows that

hkm(x) = ‖x‖khk

m(ω), ∀x =ω‖x‖ ∈Rn , where ω= x

‖x‖ ∈ Sn−1.

Hence

dn,k∑m=1

(hk

m(x))2 = ‖x‖2k , ∀x ∈Rn . (8.3)

Recall from Theorem 8.2.2 that

Rhkα

f (ξ) = mp (ξ) f (ξ) := i k hkα(ξ)

|ξ|k f (ξ).


Let k ∈ 2N−1. Then R is self adjoint, since

⟨⟨R f , g

⟩⟩= ⟨⟨R f , g

⟩⟩= dn,k∑α=1

∑β,γ∈Odn,k

eβeαeγ

∫Rn

(−1)(k−1)/2ihkα(x)

|x|k fβ(x)gγ(x)d x

=dn,k∑α=1

∑β,γ∈Odn,k

eβ(−eα)eγ

∫Rn

fβ(x)−(−1)(k−1)/2ihk

α(x)

|x|k gγ(x)d x

=dn,k∑α=1

∑β,γ∈Odn,k

eβeγeα

∫Rn

fβ(x)(−1)(k−1)/2ihk

α(x)

|x|k gγ(x)d x

= ⟨⟨f ,Rg

⟩⟩= ⟨⟨

f ,Rg⟩⟩

,

where we used Plancherel’s equality. (See Theorem 3.2.15 on page 58).

Now we need to show that⟨⟨

R f ,Rg⟩⟩= ⟨⟨

f , g⟩⟩

. Indeed

⟨⟨R f ,Rg

⟩⟩=⟨⟨RR f , g⟩⟩= dn,k∑

β,γ=1

∑α,δ∈Odn,k

eαeγeβeδ

∫Rn

−hkβ

(ξ)hkγ(ξ)

‖ξ‖2kfα(ξ)gδ(ξ)dξ

= ∑α,δ∈Odn,k

eα( ∑

1≤β<γ≤dn,k

(eγeβ−eβeγ)︸︷︷︸=0

∫Rn

−hkβ

(ξ)hkγ(ξ)

‖ξ‖2kfα(ξ)gδ(ξ)dξ

+∫Rn

dn,k∑β=1

e2β︸︷︷︸

=−1

−(hkβ

(ξ))2

‖ξ‖2k

︸︷︷︸=1, by (8.3)

fα(ξ)gδ(ξ)dξ)eδ =

⟨⟨f , g

⟩⟩

= ⟨⟨f , g

⟩⟩.

The proof for k ∈ 2N is quite similar. Since in this case i k ∈R we have to multiply by i to usethe argumentation above. This proves that iR is self adjoint. It follows that R∗ = −R. Thisyields the negative sign needed in the proof of unitarity.

Using Theorem 8.2.7, we can show that the higher Riesz transforms map Clifford frames ontoClifford frames.

Theorem 8.2.8 (Frame property of higher Riesz transforms)LET fl l∈N ⊂ L2(Rn ,R) be a frame for L2(Rn ,R) with frame bounds A and B . Furthermorelet hk

1 , . . . ,hkdn,k

⊂ Hk be a basis of Hk as in Theorem 8.2.7 and let R be the corresponding

hypercomplex higher Riesz transform.

(i) Then fl l∈N is a Clifford frame for L2(Rn ,R)dn,k∼= L2(Rn ,Rdn,k

) with the same framebounds A and B .

(ii) R fl l∈N is a Clifford frame for L2(Rn ,R)dn,kwith the same frame bounds A and B .

Proof. The proof follows directly from Theorem 5.2.13 on page 86 just as the proof of Theo-rem 5.3.3.


8.2.3 A phase concept for higher Riesz transforms

In Definition 8.2.4 we have established hypercomplex higher Riesz transforms. In this sectionwe define a higher monogenic signal to obtain a phase amplitude decomposition of a signal.Furthermore we present a kind of Cauchy-Riemann equation which is satisfied by the Pois-son transform of the higher monogenic signal just as the Poisson transform of the monogenicsignal satisfies the Cauchy Riemann equations (3.4). (See Theorem 4.5.2 on page 70. )

The first step is to define a phase direction for the higher Riesz transform of a function.

Definition 8.2.9 (Phase direction for higher Riesz transforms)

LET f ∈ L2(Rn) and let(Rhk

l

)dn,k

k=1 ∈ L2(Rn)dn,kbe its higher Riesz transform.

THEN the phase direction d is defined as

d :Rn →Rdn,k , d = Rk f

|Rk f | =(Rhk

lf)dn,k

k=1

|Rk f | .

Remark 8.2.10 (Interpretation of the phase direction)The phase direction of a higher Riesz transform is at first view hard to interpret, since the do-main of the signal is n-dimensional while the phase direction is dn,k dimensional. It seems

reasonable to interpret the phase by the spherical harmonic d = ∑dn,k

l=1 dlhkl . In the case of the

Riesz transforms, that is in the case that k = 1, this polynomial has exactly one preferred direc-tion - the phase direction of the Riesz transform.

For n = 2 the higher Riesz transform has exactly k preferred directions which are equally spaced.It can easily be seen in the explicit formulas in Theorem 8.3.1 and the corresponding Figure 8.1that directions which defer by a multiple of π

k can not be distinguished. This results in a direc-tional ambiguity.

In the case n = 3, k > 1 a finer distinction is possible - there exist spherical harmonics with ex-actly one preferred direction and spherical harmonics with more then one preferred direction.(See Figure 8.2 and Figure 8.3.)

Following the definition of monogenic phase in section 4.4 we can establish phase and ampli-tude for higher Riesz transforms:

Definition 8.2.11 (Amplitude - phase decomposition of higher Riesz transforms)LET f ∈ L2(Rn).

THEN the higher monogenic signal is defined as

fm,k := f +Rk f ∈ L2(Rn ,Rdn,k+1).

We can decompose fm,k in the same way as the monogenic signal:

fm,k = ∣∣ fm,k∣∣( f

| fm,k |+ Rk f

|Rk f ||Rk f || fm,k |

)= a

(cos

(φ

)+d sin(φ

))= a exp(dφ

), (8.4)

where a = | fm,k | is called the amplitude, φ = arg(

f| fm,k | + i |Rk f |

| fm,k |)

is called the phase and

d = Rk f|Rk f | is called the phase direction of the signal f .


Remark 8.2.12 (Higher Riesz transforms of wavelets)Letψ ∈ L2(Rn) such thatψ is a mother wavelet for a wavelet frame D j Ttψ j∈Z,t∈Zn of L2(Rn).Let f ∈ L2(Rn).

• Then by Theorem 8.2.8 it follows that D j Tt Rψ j∈Z,t∈Zn is a multi wavelet frame ofL2(Rn). By Example 8.2.1 3. this frame is steerable.

• The phase direction defined in Definition 8.2.9, amplitude and phase defined in Defini-tion 8.2.11 are defined in the same way for the coefficients of this wavelet decomposition.

Wψm,k ( f ) = ⟨⟨f ,D j Ttψm,k

⟩⟩j∈Z,t∈Zn =

a j ,t exp(d j ,tφ j ,t

)j∈Z,t∈Zn .

• Let ψ ∈H0. Then Rhklψ

dn,k

l=1 is a basis for a linear subspace of Hk . The phase direction

of the wavelet transform corresponds to the element

Rd j ,tψ⊂ spanRhklψ, l = 1, . . . ,dn,k ⊂Hk

of norm ‖Rd j ,tψ‖ = ‖ψ‖ that has the highest scalar product with T−t D− j f .

That is we approximate the signal f with the optimal element Rd j ,tψ in the subspacespanRhk

lψl=1,...,dn,k

ofHk (Rn). To find this element we need the higher Riesz transform

which consists of a basis of this subspace.

To state the Cauchy Riemann equation that the Poisson transform of the higher Riesz trans-forms satisfy, we first define the Cauchy-Riemann operators whose kernel satisfies the gener-alized Cauchy Riemann equations.

Definition 8.2.13 (Higher Dirac operator)

LET f ∈W k,2(Rn). Furthermore, let hkl

dn,k

l=1 be a ONB of Hk . Then

hkl = ∑

β∈Nn0

|β|≤k

r kl ,βxβ,

for some r kl ,β ∈C.

THEN the higher Dirac operator of order k is defined as

Dk : W k2 (Rn ,Rdn,k

) → L2(Rn ,Rdn,k), f 7→

dn,k∑α=1

eαd−1/2n,k hk

α(∂) f ,

where hkα(∂) :=∑

|β|≤k r kβ,α∂

β =∑|β|≤k r k

β,α∂β1

∂xβ11

· · · ∂βn

∂xβnn

.

The corresponding higher Cauchy-Riemann operator is

∂k := ∂k

∂xk0

+Dk : W k,2(Rn+1,Rdn,k) → L2(Rn+1,Rdn,k

).

The conjugate Cauchy-Riemann operator is ∂k := ∂k

∂xk0−Dk


Corollary 8.2.14 (Factorization of powers of the Laplacian)

LET 4=∑dn,kα=1∂

2α : W 2,2(Rn) → L2(Rn) be the Laplacian and let

4k : W 2k,2(Rn) → L2(Rn), f 7→ (k∏

l=14) f

denote its k-th power.

THEN

D2k =−4k .

Furthermore

∂k∂k = ∂2k

∂x2k0

+4k .

Proof. Let f ∈W 2k,2(Rn).

F (D2k f )(ξ) =F

(dn,k∑α=1

dn,k∑β=1

eαeβd−1n,kh

kα(∂)hk

β(∂) f

)(ξ) = d−1

n,k

dn,k∑α=1

dn,k∑β=1

eαeβhkα(−2πiξ)hk

β(−2πiξ) f (ξ)

= (−2πi )2k d−1n,k

(−

dn,k∑α=1

(hkα(ξ)

)2 + ∑α,β=1,...,dn,k

α<β

eαeβhkα(ξ)hk

β(ξ)+eβeαhkβ(ξ)hk

α(ξ))

f (ξ)

= (−2πi )2k d−1n,k

(−dn,k |ξ|2k + ∑

α,β=1,...,dn,kα<β

eαeβ(hkα(ξ)hk

β(ξ)− (ξ)hkα(ξ)hk

β

)︸︷︷︸=0

)f (ξ)

=−F(4k f

)(ξ), f.a.a. ξ ∈Rn .

∂k∂k f (x) = ∂2

∂x20

f (x)+ ( ∂

∂x0Dk −Dk

∂

∂x0

)f (x)−D2

k f (x), ∀ f ∈W 2k,2(Rn+1), f.a.a. x ∈R.

Theorem 8.2.15LET

f =dn,k∑α=0

eα fα ∈ Lp (Rn ,Rdn,k+1+ ), 1 < p <∞, if k is odd,

and

f = e0 f0 − idn,k∑α=1

eα fα ∈ Lp (Rn ,Rdn,k+1+ ), 1 < p <∞, if k is even.

Furthermore, let

uα(x, x0) := Px0 ∗ fα(x), ∀α= 0, . . . ,dn,k , x0 ∈R+, x ∈Rn .

THEN fα =Rhkα

( f0), α= 1, . . . ,dn,k , iff u satisfies

∂k u = 0, if k is odd, and( ∂k

∂xk0

+ i D)u = 0, if k is even.


This is equivalent to the following set of generalized Cauchy Riemann equations:

hkβ(D)uα(x)−hk

α(D)uβ(x) = 0, ∀α,β= 1, . . . ,dn,k ; (8.5)

∂k uα

∂xk0

(x)+ (−1)k+1hkα(D)u0(x) = 0, ∀α= 1, . . . ,dn,k ; (8.6)

n∑α=1

hkα(D)uα(x)− ∂k u0

∂xk0

(x) = 0, f.a.a. x0 ∈R+, x ∈Rn . (8.7)

Proof. The equalities in this proof hold for almost all x ∈Rn , x0 ∈R+.

If k is odd, then

∂k u(x, x0) = ∂k

∂xk0

u0(x, x0)−dn,k∑α=1

hkα(D)uα(x, x0)

+dn,k∑α=1

eα(hkα(D)u0(x, x0)+ ∂k

∂xk0

uα(x, x0))

+dn,k∑

α6=β=1eαeβ

(hkα(D)uβ(x, x0)−hk

β(D)uα(x, x0))).

This equals 0 iff the generalized Cauchy Riemann equations (8.5), (8.6) and (8.7) hold.

If k is even, then

( ∂k

∂xk0

+ i D)u(x, x0) = ∂k

∂xk0

u0(x, x0)− i (−i )dn,k∑α=1

hkα(D)uα(x, x0)

+dn,k∑α=1

eα(ihk

α(D)u0(x, x0)− i∂k

∂xk0

uα(x, x0))

+dn,k∑

α6=β=1i (−i )eαeβ

(hkα(D)uβ(x, x0)−hk

β(D)uα(x, x0))).

Which is equal to 0 iff the generalized Cauchy Riemann (8.5), (8.6) and (8.7) equations hold.

Suppose fα = Rhkα

f0, then fα(t ) = i khkα(t )

|t |k f0(t ) and, hence,

uα(x, x0) =∫Rn

f0(t )hkα(i t )

|t |k e−2π|t |x0 e2πi t x d t , ∀α= 1, . . . ,dn,k ,

and

u0(x, x0) =∫Rn

f0(t )e−2π|t |x0 e2πi t x d t .

By the dominated convergence theorem [41], we may differentiate under the integral sign, and


hence obtain

∂k u0

∂xk0

(x, x0) = (−2π)k∫Rn

f0(t )|t |k e−2π|t |x0 e2πi t x d t

hkβ(∂)u0(x, x0) =

∫Rn

f0(t )(2πi )khkβ(t )e−2π|t |x0 e2πi t x d t

∂k uβ

∂k x0(x, x0) =

∫Rn

f0(t )(−2π)k i khkβ(t )e−2π|t |x0 e2πi t x d t

hkβ(∂)∂uα(x, x0) =

∫Rn

f0(t )(2πi )k i khkα(t )hk

β(t )

|t |k e−2π|t |x0 e2πi t x d t

Now (3.4) is easy to check. The first equation (8.5) follows from

hkβ(∂)∂uα(x, x0) = (−2π)k

∫Rn

f (t )hkα(t )hk

β(t )


= hkα(∂)uβ(x, x0).

The second identity (8.6) follows from

hkβ(∂)∂u0(x, x0) = (2πi )k

∫Rn

f (t )hkβ(t )e−2π|t |x0 e2πi t x d t

= (−1)k ∂k uβ

∂k x0(x, x0).

Finally the last identity (8.7) follows from

dn,k∑α=1

hkα(∂)uα(x, x0) = (−1)k

dn,k∑α=1

(2π)k∫Rn

f (t )

(hkα(t )

)2


= (−2π)k∫Rn

f (t )

∑dn,kα=1

(hkα(t )

)2


= (−2π)k∫Rn

f (t )|t |k e−2π|t |x0 e2πi t x d t

= ∂u

∂x0(x, x0).

Conversely, let β ∈ 1, . . . ,dn,k and uβ(x, x0) = ∫Rn fβ(t )e−2π|t |x0 e2πi t x d t . The fact that

hkβ

(∂)u0 = (−1)k ∂k uβ∂k x0

, shows that

(2πi )k∫Rn

f (t )hkβ(t )e−2π|t |x0 e2πi t x d t = 2π

∫Rn

fβ(t )|t |k e−2π|t |x0 e2πi t x d t .

Therefore fβ(t ) = i khkβ

(t )

|t |k f (t ), and so

fβ =Rhkβ

( f ), β= 1, . . . ,dn,k .


8.2.4 Composition of higher Riesz transforms

As is easily seen by Figure 8.1, Figure 8.2 and Figure 8.3 the higher Riesz transforms basedon spherical harmonics are restricted with respect to their geometry. It might for example

be interesting to look at the higher Riesz transforms given by the Fourier multiplier(

i xl|x|

)k.

However if k ≥ 2 the polynomial xkl |Sn−1 lies in Pk |Sn−1 = ∑bk/2c

l=0 Hk−2l but not in Hk . In thissection we derive higher Riesz transforms based on polynomial spaces composed of sphericalharmonics.

The higher Riesz transforms are a vector consisting of the basis elements of a rotation invariantvector space

RHk:= spanRhk

l, l = 1, . . . ,dn,k

of operators on Lp (Rn), 1 < p <∞. This vector space has an irreducible representation of therotation group equivalent to the Wigner-D matrices. We can compose these spaces to gainnew spaces of operators that feature a reducible representation of the rotation group. In factall such operator spaces that are compatible with wavelets can be composed of such spaces ofhigher Riesz transforms.

The proof that an operator T on a certain Clifford-Hilbert module maps frames onto framesis based on Theorem 5.2.13 and depends only on the fact that the operator T is invertible. Iffurthermore ‖T †‖ = ‖T ‖ = 1, where T † is the pseudo-inverse of T , then T maps tight framesonto tight frames. In Theorem 8.2.8 we chose the higher Riesz transforms Rk to be unitary.The proof of the unitarity Theorem 8.2.7 depends only on (8.2), namely that

dn,k∑l=1

(d−1/2

n,k hkl (ω)

)2 = 1, ∀ω ∈ Sn−1.

Furthermore, the higher Riesz transforms satisfy R−1k =R∗

k = (−1)kRk .

Definition 8.2.16Let K = k1, . . . ,kk ⊂N0, where k ∈N and kl 6= km ,∀l 6= m ∈ K . We define a higher Riesz trans-form that consists of basis elements of the spaces RHkr

, r ∈ K . Let

RK : L2(Rn)∑k∈K dn,k

→ L2(Rn)∑k∈K dn,k

,

RK := ∑k=k1,...,kk

dn,k∑l=1

eαk,l dK Rhkl

, (8.8)

where αk,l =∑k−1

m=1 dn,m + l , and dK ∈C is chosen such that

∑k∈K

dn,k∑l=1

(dK h

kl (ω)

)2 = 1, ∀ω ∈ Sn−1. (8.9)

Corollary 8.2.17(8.9) is easily satisfied by setting

dK = ( ∑k∈K

dn,k)−1/2.

Proof. Indeed is true that∑

k∈K∑dn,k

l=1

(dK h

kl

)2 = d 2K

∑k∈K dn,k = 1.


Corollary 8.2.18 (Composition of higher Riesz transforms)Let K as above. Then R⊕

k∈K Hk= ⊕

k∈K RHkis a rotation invariant vector space of linear

bounded operators in Lp (Rn) invariant under translation and dilation. Its elements are uniquelydefined by a vector a = (am

l )l=1,...,dn,kk∈K

, akl ∈C, via

Ra = ∑k∈K

dn,k∑l=1

akl Rhk

l.

A reducible representation S of the rotation group on R⊕k∈K Hk

is given by

ρ−1 Ra ρ =SρRa := ∑k∈K

dn,k∑l=1

(Dkρ ak )l Rhk

l,

where Dlρ is the Wigner-D matrix and ak ∈Cdn,k = (ak

1 , . . . , akdn,k

).

A higher Riesz transform can be defined as a vector of basis elements of R⊕k∈K Hk

the canonicalexample is

RK := (dK Rhk

l

)l=1,...,dn,k

k∈K

.

This higher Riesz transform is steerable because of the representation S of the rotation groupon R⊕

k∈K Hk.

The adjoint of the hypercomplex higher Riesz transform (8.8) is

R∗K = ∑

k=k1,...,kk

dn,k∑l=1

(−1)k eαk,l dK Rhkl

.

The same argumentation as in Theorem 8.2.7 yields that RK is a unitary operator, whence it isbijective with inverse R∗ and hence using Theorem 5.2.13 on page 86 it maps frames of L2(Rn)to multiframes of L2(Rn).

EXAMPLE 8.2.2 (Higher Riesz transforms for the spaces Pk ):

Let K = k −2m

bk/2cm=0 . Then R⊕

k∈K Hk= RPk

. Indeed, let p ∈Pk . Theorem 8.1.4 implies that

Pk =⊕b k2 c

m=0 | · |2mHk−2m . As a consequence there exist ak−2ml , l = 1, . . . ,dn,k−2m , m = 0, . . . ,

⌊ k2

⌋such that

p(x) =b k

2 c∑m=0

dn,k−2m∑l=1

ak−2ml hk−2m

l (x)|x|2m , ∀x ∈Rn .

Let f ∈ L2(Rn). Then

Rp f (ξ) = i k p(ξ)

|ξ|k f (ξ) =i k ∑b k

2 cm=0

∑dn,k−2m

l=1 ak−2ml hk−2m

l (x)|ξ|2m

|ξ|k f (ξ)

b k2 c∑

m=0

dn,k−2m∑l=1

i 2mi k−2m ak−2m

l hk−2ml (ξ)|ξ|2m

|ξ|k f (ξ)

=b k

2 c∑m=0

dn,k−2m∑l=1

i 2mi k−2m ak−2m

l hk−2ml (ξ)

|ξ|k−2mf (ξ)

=b k

2 c∑m=0

dn,k−2m∑l=1

(−1)m ak−2ml

áRhk−2ml

f (ξ).


That is we can write any partial higher Riesz transform with respect to a homogenous poly-nomial of order k as a linear combination of the the partial higher Riesz transforms of orderk −2m, m = 0, . . . ,b k

2 c.

Let RPk:= spanRp : p ∈Pk . Then RPk−2

⊂RPk−2⊕RHk =RPk

.

In view of Remark 8.2.3 this inclusion is obvious, since on the unit circle the polynomialp ∈Pk (Rn) and the polynomial | · |2l p ∈Pk+2l (Rn) agree. As a consequence the space RPk

is the space of higher Riesz transforms with respect to polynomials of degrees k −2m, wherem = 0, . . . ,b k

2 c.

Let d k be the phase direction corresponding to the higher Riesz transform of order k.

Then d(x) :=∑dn,k

l=1 d kl h

kl (x)−∑dn,k−2

l=1 d k−2l ‖x‖2hk−2

l (x) ∈Pk

d(x) :=b k

2 c∑m=0

dn,k−2m∑l=1

(−1)md k−2ml hk−2m

l (x)‖x‖2m

defines a phase direction in the homogenous polynomials of degree k.

Remark 8.2.19 (Alternative bases for R⊕ml=1Hkl

)

Let us state a method to generate alternative bases of R⊕ml=1Hkl

from the standard basis which

is given by Rhkl

l=1,...,dn,k ,k∈K . For some a = (akl )l=1,...,dn,k ;k∈K , ak

l ∈C, let

a ·RK := ∑k∈K

dn,k∑l=1

akl dK Rhk

l.

The basis vectors of the standard basis can be written as Rhkl= a(l ,k) ·RK , where

a(l ,k) = (ar

m(l ,k))

m=1,...,dn,k ,r∈K = (δl ,mδk,r)

m=1,...,dn,k ,r∈K ∈R∑

k∈K dn,k .

Let A ∈ U(∑

k∈K dn,k ) be a orthogonal matrix onR∑

k∈K dn,k . It is a simple fact from linear algebrathat A maps the orthonormal basis

dK Rh(l ,k)

l ,k to an orthonormal basis

(Aa)k

l ·RK

l ,k .

Let B ∈ Gl(∑

k∈K dn,k ) be a invertible matrix with inverse B−1. Then(B a)k

l ·RK

l ,k is a basis for

the space HK . Let ψr r ⊂ L2(Rn) be a frame and let φr r ⊂ L2(Rn) be the dual frame. Thenthe dual frame of the frame

(B a)kl ·RKψr

r ;l=1,...,dn,k ,k∈K

is the frame (B−1((−1) j a(m, j )

)m=1,...,dn, j ; j∈K

)k

l·RKψr

r ;l=1,...,dn,k ,k∈K

.

Up to now we have shown how to combine a set of higher Riesz transforms based on spheri-cal harmonics into a new higher Riesz transform and how to change the basis elements of thehigher Riesz transform. Next we show how to do it the other way round: We chose a polyno-mial and build a basis for the minimal space of higher Riesz transforms such that the partialhigher Riesz transform with respect to the chosen polynomial is an element of the higher Riesztransform.

Remark 8.2.20 (Higher Riesz transforms based on polynomials)Inspired by the steerable filters of Freeman and Adelson in [18] we want to state another, more


constructive method to construct a steerable set of operators on L2(Rn) based on a single op-erator. For this purpose let p be a polynomial of degree k ∈N. We assume that p is normalizedsuch that ∫

Sn−1|p(x)|2dσ(x) = 1.

Then there exist a set K ⊂N and (pkl )l ,k ∈R

∑k∈K dn,k such that

p = ∑k∈K

pkl h

kl ∈ ⊕

k∈KHk .

By our assumption on the normalization of p we know that∣∣(pkl )l ,k

∣∣C

∑k∈K dn,k = 1.

A basis RB(l )l=1,...,∑

k∈K dn,kof RK is constructed by setting RB1 = Rp and extending with el-

ements of the standard basis Rhkl

l ,k or a linearly independent set of rotated versions of Rp .

This basis can then be converted to an orthonormal basis of RK by applying the Gram-Schmidtprocess to the coefficients of the basis RB(l )l=1,...,

∑k∈K dn,k

with respect to the standard basisRhk

ll ,k .

The approach in Remark 8.2.20 is especially useful since the operators Rp are similar to deriva-tions as stated in the following.

Theorem 8.2.21 (Higher Riesz transforms and derivatives)LET f ∈ L2(Rn) such that f = 4k/2g for some g ∈ W k,2(Rn) and let p ∈ Pk . (Here 4k/2 :W k,2(Rn ) → L2(Rn) is the k-th order (fractional) differential operator given by the Fourier sym-bol (2πi )k |ξ|k .)

THEN the partial Riesz transform Rp can be interpreted as an derivative in the sense that

Rp f = (i )k p(∂)g .

Proof.

F(Rp f

)(ξ) = p(iξ)

|ξ|k f (ξ) = p(iξ)(2πi )k |ξ|k

|ξ|k g (ξ) = (i )kF(p(∂)g

)(ξ), f.a.a. x ∈Rn .

We now show that steerable pyramids are based on partial higher Riesz transforms:

Remark 8.2.22 (Steerable filters and higher Riesz transforms)To construct a steerable pyramid in 2-D the proposed filter f : R2 → R is written in polarcoordinates and then decomposed into a Fourier series with respect to the angular coordinate:

f (r,φ) =k∑

l=−kal (r )e2πi lφ. (8.10)

The minimum number of filters needed for steering is the number m of indices l for which∃r ∈R+

0 : al (r ) 6= 0. As a consequence a filter is steerable iff the highest order k <∞. A basisfor the rotation invariant space in which the filter f lies is then given by ρl f m−1

l=0 , where ρl isa rotation by the angle 2π/m.


Let us give an interpretation for this approach. Let f ∈ L2(R2,R) and suppose that

∃ f :R+0 →R : al (r ) = f (r )al (1). (8.11)

Remember that e−2πi k · ,e2πi k · is a basis for Hk . Thus (8.10) states that

f ∈ H f := ⊕k∈N:ak (r )6=0

Hk .

If (8.11) does not hold then ∀l∃ fl (r ) : R+0 → R : al (r ) = fl (r )al (1) and hence f ∈ H f . Note

that since f is real valued al 6= 0 ⇔ a−l 6= 0. As a consequence m is the dimension of H f .Since H f is rotation invariant, rotating f yields an element of H f . Let ρl be a rotation by2πlm . To show that the m rotated versions of f yield a basis for the rotation invariant subspace

spanρl f , l = 0, . . . ,m ⊂ H f it only remains to show that they are linearly independent - thisholds true since the spaces Hk are irreducible and orthogonal to each other. If (8.11) holds byCorollary 8.1.12 the filter f is the partial higher Riesz transform with respect to the polynomialp = ∑

l al e2πi l · of the function f . The rotated filters are partial higher Riesz transforms of fwith respect to rotated versions of the polynomial p.

If Equation 8.11 does not hold, then the signal f is the sum of the partial higher Riesz trans-forms of the function fl with respect to the polynomials al e−2πi l ( · ).

8.3 Implementational aspects

In this section we discuss the implementation of higher Riesz transforms. The first step to-wards an implementation is of course the construction of anorthonormal bases of the spacesHk (Rn) and the corresponding steering matrices D. We discuss such constructions for theinteresting cases n = 2,3 for which there is a proper physical interpretation for rotations. Thenext step is the implementation of the higher Riesz transforms via wavelet frames. This is dis-cussed in subsection 8.3.2.

8.3.1 Explicit constructions for L2(R2) and L2(R3)

In this section we will give some explicit constructions of orthonormal bases of the spacesHk (Rn) for n = 2,3.

A basis for Hk (R2)

Theorem 8.3.1LET k ∈N.

THEN d2,k = 2. A basis for Hk (R2) is given by

yk1 , yk

2 = ℜ((Y (x))k ),ℑ((Y (x))k ),

where Y (x) := x1 + i x2, ∀x = (x1, x2) ∈ S1. Furthermore, ⟨ f , g ⟩ =⟨

yk1

|x|k f ,yk

1

|x|k g⟩+

⟨yk

2

|x|k f ,yk

2

|x|k g⟩

.

A rotation by an angle φ ∈ [0,2π] is given by the matrix

Dρ,k =(

yk1 (cos(φ),sin(φ)) −yk

2 (cos(φ),sin(φ))yk

2 (cos(φ),sin(φ)) yk1 (cos(φ),sin(φ))

).

8.3. IMPLEMENTATIONAL ASPECTS 157

y k1 y k

2k = 0 y0

1 (x1, x2) = 1 y02 (x1, x2) = 0

k = 1y1

1 (x1, x2) = x1 y12 (x1, x2) = x2

k = 2y2

1 (x1, x2) = x21 −x2

2 y22 (x1, x2) = 2x1x2

k = 3y3

1 (x1, x2) = x31 −3x1x2

2 y32 (x1, x2) = 3x2x2

1 −x32

k = 4y4

1 (x1, x2) = x42 −6x2

1 x22 +x4

1 y42 (x1, x2) = 4x3

1 x2 −4x32 x1

Figure 8.1: The basis elements given in Example 8.3.1 of the spherical harmonics for n = 2,k = 0, . . . ,4.

Proof. The operators are obviously self-adjoint. Hence we have to proof that(yk

1 (x)

|x|k)2

+(

yk2 (x)

|x|k)2

= 1.


This is implied by the addition theorem 8.1.8 or easily proven by observing that(yk

1 (x)

|x|k)2

+(

yk2 (x)

|x|k)2

= |x|−k((ℜ(x1 + i x2)k)2 + (ℑ(x1 + i x2)k)2

)= |x|−k (x1 + i x2)k (x1 + i x2)k = |x|−k∥∥(x1 + i x2)k∥∥= 1.

A rotation of imaginary numbers by an angle φ is achieved by multiplication by e iφ. Hence

Dρ,k (yk1 , yk

2 ) =(ℜ(

e i kφ(x1 + i x2)k),ℑ(

e i kφ(x1 + i x2)k)).

Now, the conclusion follows, since

ℜ(e i kφ(x1 + i x2)k)= yk

1

(cos(φ),sin(φ)yk

2 (x1, x2)− yk2

(cos(φ),sin(φ)yk

2 (x1, x2))

and

ℑ(e i kφ(x1 + i x2)k)= yk

2

(cos(φ),sin(φ)yk

1 (x1, x2)+ yk1

(cos(φ),sin(φ)yk

2 (x1, x2)).

EXAMPLE 8.3.1:

k = 0 : y01 = 1, y0

2 = 0

k = 1 : y11 = x1, y1

2 = x2

k = 2 : y21 = x2

1 −x22 , y2

2 = 2x1x2

k = 3 : y31 = x3

1 −3x22 x1, y3

2 = 3x21 x2 −x3

2

k ∈N : Using Tchebichef polynomials we write

yk1 = 1

|x|kb k

2 c∑j=0

(−1) j

(k

2 j

)x2 j

1 xk−2 j2 , yk

2 = 1

|x|kd k

2 e∑j=0

(−1) j+1

(k

2 j −1

)x2 j−1

2 xk−2 j+11

The resulting Higher Riesz transforms are as follows:

The identity

k = 0 Ry01

f = f , Ry02

f = 0,

the Riesz transforms

k = 1 Ry11

f (ξ) = iξ1

|ξ| f (ξ), Ry12

f (ξ) = iξ2

|ξ| f (ξ),

and the higher Riesz transforms

k ∈N Ryk1

f (ξ) = (−i |ξ|)−k ykc f (ξ), Ryk

2f (ξ) = (−i |ξ|)−k yk

s f (ξ).


A basis for H k (R3)

Normalized spherical harmonics

Definition 8.3.2Let the 2k +1 associated Legendre polynomials of degree k ∈N be given by

P mk (x) := (1−x2)m/2 d k+m

d xk+m(x2 −1)k ,∀x ∈R,

where m =−k, . . . ,k. Let (r,θ,φ) ∈R3 be given in spherical coordinates – see Appendix A.1.2. Anypoint on the unit sphere S2 is then determined by two coordinates (θ,φ) = (1,θ,φ).

THEN a basis for the 2k+1 dimensional space Hk (S2) of spherical harmonics of degree k is givenby Y m

k m=−k,...,k , where

Y mk (θ,φ) :=

√(k −m)!

(k +m)!(−1)me i mφP m

k (cos(θ)), ∀(θ,φ) ∈ S2.

A real valued basis of spherical harmonics y0k ∪ ym

k,c m=1,...,k ∪ ymk,s m=1,...,k is established by

setting

y0k = Y 0

k (θ,φ) = P 0k

ymk,c = 2−1/2(Y m

k (θ,φ)+ (−1)mY −mk (θ,φ)) =

√2

(k −m)!

(k +m)!(−1)m cos(mφ)P m

k (cos(θ))

ymk,s = (−i )2−1/2(Y m

k (θ,φ)− (−1)mY −mk (θ,φ)) =

√2

(k −m)!

(k +m)!(−1)m sin(mφ)P m

k (cos(θ)),

∀m = 1, . . . ,k, (θ,φ) ∈ S2.

Wigner D-matrices for R3 In R2 a rotation is described by a rotation angle. In R3 severaldifferent approaches to describe rotations are common usage. We will consider the construc-tion of the Wigner matrix in the case that the rotation is given in terms of Euler angles. Thecase that a rotation is given by an eigenvector and a rotation angle can be found in [14].

The Wigner D-matrix of a rotation given in Euler angles Any rotation ρ ∈ SO(3) can be ex-pressed as a product of three rotations which leave the z, respectively, the x-axis fixed:

ρ = ρα,zρβ,xργ,z =cos(α) −sin(α) 0

sin(α) cos(α) 00 0 1

1 0 00 cos(β) −sin(β)0 sin(β) cos(β)

cos(γ) −sin(γ) 0sin(γ) cos(γ) 0

0 0 1

The irreducible representation of the rotation group is given by the Wigner-D matrices [57]:

Theorem 8.3.3 (Explicit form of the Wigner-D matrix)LET ρ ∈ SO(3) given in Euler angels: ρ = ραρβργ.

THEN

Y km(ρ(θ,φ)) =

k∑l=−k

(D(α,β,γ),k

)m,l

Y kl (θ,φ), ∀(θ,φ) ∈ S2,


where

(D(α,β,γ),k )m,l =∑

r(−1)r ck,l ,me i mα cos2k+l−m−2r (β/2)sin2r+m−l (β/2)e i lγ

(k −m − r )!(k + l − r )!r !(r +m − l )!, (8.12)

and ck,l ,m =

(−1)l+m(k −m)!(k + l )! l ,m ≥ 0,

(k +m)!(k − l )! l ,m < 0,

(−1)l (k +m)!(k + l )! l ≥ 0,m < 0,

(−1)m(k −m)!(k − l )! l < 0,m ≥ 0.

For β= γ= 0 it follows thatD(α,0,0),k = diag(e i (m−k)α).

Furthermore D(0,0,γ),k =D(γ,0,0),k .

The Wigner-D matrix is applied to real valued spherical harmonics in the following way:

y0k =

k∑l=1

p2(

y lk,c

(D0,l +(−1)l D0,−l

)+ i y lk,s

(D0,l −(−1)l D0,−l

))+D0,0 y0k ;

ymk,c =

k∑l=1

y lk,c

(Dm,l +(−1)l Dm,−l +(−1)m D−m,l +(−1)m+l D−m,−l

)+ i y l

k,s

(Dm,l −(−1)l Dm,−l +(−1)m D−m,l −(−1)l+m D−m,−l

)+2−1/2 y0

k

(Dm,0+(−1)m D−m,0

);

ymk,s =

k∑l=1

i y lk,c

(Dm,l +(−1)l Dm,−l −(−1)m D−m,l −(−1)l+m D−m,−l

)− y l

k,s

(Dm,l −(−1)l Dm,−l −(−1)m D−m,l +(−1)l+m D−m,−l

)+ i 2−1/2 y0

k

(Dm,0−(−1)m D−m,0

).

8.3.2 Implementational aspects

According to Theorem 8.2.8 the higher Riesz transform of a wavelet frame is a wavelet frame forL2(Rn)dn,k

. Hence we can use the same approach for the implementation of the higher Riesztransforms that we used for the Riesz transform in subsection 5.6.1 – an implementation viathe isotropic wavelets constructed in Example 5.5.1. Once again the filters are implementedas perfect reconstruction filter bank in the frequency domain since they have explicit formulasand bounded support in the frequency domain. The Shannon sampling theorem ensures loss-less down-sampling after each filter step. As we constructed tight wavelet frames, the synthesisfilter bank consists of the same filters as the analysis filter bank.

The Fourier multipliers of the higher Riesz transforms share a singularity at ξ= 0 (DC compo-nent). We deal with this singularity by subtracting the images mean value beforehand, whichsets the DC component equal to zero.

The runtime of the higher Riesz wavelet decomposition is determined by that of the fastFourier transform, namely O(N log N ). The memory consumption amounts to a factor of lessthan dn,k times the image size, where n = 2 or n = 3 is the image dimension and k the degreeof the higher Riesz transform.


Note that in practice the order k of the higher Riesz transforms is limited by the angular reso-lution of the filters - due to the discrete nature of the filters only a finite number of directionscan be resolved.

8.3.3 Discussion

The present chapter gives a method to construct linear bounded operators - the higher Riesztransforms - which are invariant under translation and dilation and map wavelet frames tosteerable multiwavelet frames. The invariance of the higher Riesz transforms under dilationand translation means that the dual frames are again multiwavelet frames.

The steerabilty is the manifestation of the irreducible representations of order k of the rotationgroup - the elements of the higher Riesz transforms are the basis elements of a linear rota-tion invariant space RHk

of linear bounded operators invariant under dilation and translationon which the representation acts. Since the representation is irreducible these spaces are theminimal spaces with these properties. As a consequence any such rotation invariant space V ofsteerable linear bounded operators invariant under dilation and translation can be composedof the spaces RHk

. That is

∃k1, . . . ,kl ⊂N, l ∈N : V =l⊕

r=1RHkr

.

Furthermore, a phase direction is defined on the spaces RHkand on the component spaces of

either only odd or only even order. Applied to the wavelet decomposition the phase directionyields a local estimate of the geometry of a signal.

8.3.4 Comparison with other approaches

In [52],[51] Michael Unser and Dimitri Van De Ville gave a definition of higher order Riesztransforms that is closely related - the elements of their higher order Riesz transforms are abasis for RPk

. Our approach is much more flexible and with the theoretical background wehave set up we can give new insights into their work.

The flexibility of our work stems from the possibility to compose the elementary spaces RHk

in an arbitrary way thus controlling symmetry and redundancy. In contrast the spaces RPk

come in two sets of even or odd order. Within these sets the spaces of lower order are includedin the spaces of higher order - it is not possible to choose one order without using all lowerorders. However this is not apparent in the basis Unser et al. chose for these spaces. In contrastthe basis we present is composed of sets of bases of the minimal rotation invariant subspacesand thus does allow to analyze the behavior of the signal in these subspaces as well as on thewhole space. Furthermore the higher Riesz transforms we defined allow the computation ofthe phase direction and phase decomposition for the whole space as well as for the rotationinvariant subspaces.

The theoretical background given in this chapter can explain some phenomenon remarkedby Unser et al. The orthogonality of the steering matrices and their group structure remarkedabout in [53] is caused by the unitary representation of the rotation group.

In [53] Unser et al. propose to apply a unitary matrix pointwise to the higher order Riesz trans-forms. This is of course just a change of the basis of the space RPk

.

The implementation of the higher Riesz transformed wavelets is a steerable pyramid as definedby Freeman and Adelson in [18]. This is however not due to chance but due to the fact that the


steerable filters designed by Freeman and Adelson are the basis of a certain rotation invari-ant subspace of functions. Indeed these basis elements consist of higher Riesz transforms ofsuitable functions.

The main difference between higher Riesz transforms and steerable pyramids is that steerablepyramids design one filter that will be a basis element of the rotation invariant space and thendesign a basis of rotated versions of this filter of the appropriate minimal rotation invariantspace. In contrast we chose the rotation invariant spaces and then find an appropriate basisfor them. Adaptivity is in our approach not part of the choice of basis - rather the phase direc-tion of the wavelet decomposition choses the wavelet frame that is locally best adapted to thesignal.

What we have found in Remark 8.2.22 is that the steerable filters of Freeman and Adelson arehigher Riesz transforms. The reason why these filters work only in discrete space is the con-struction of the radial function f - they are constructed in a way to yield a filter bank. Of courseit is not guaranteed that this filter corresponds to a wavelet frame of L2(Rn).


ymkc ym

ksk = 0

m = 0y0

0 = 1k = 1

m = 0y0

1 = cos(θ)sin(φ)

m = 1y1

1c = cos(φ)cos(θ) y11s = sin(θ)

Figure 8.2: The basis elements given in Definition 8.3.2 of the spherical harmonics for n = 3,k = 0,1.


ymkc ym

ksk = 2

m = 0

y02 = 3cos2(Θ)−1

2

m = 1

y12,c =

p6sin(2φ)cos(θ)

23/2 y12,s =

p6sin(2φ)sin(θ)

23/2

m = 2

y22,c =

p6(

sin(2φ)cos(2θ)+sin2(φ)−cos2(φ))

25/2 y22,s =

p6(

sin(2φ)cos(2θ)−2cos(φ)sin(φ))

25/2

Figure 8.3: The basis elements given in Definition 8.3.2 of the spherical harmonics for n = 3,k = 2.

Appendix A

The Appendix

A.1 Conventions and definitions

A.1.1 Lie groups and Lie algebras

LET L be a vector space over K, where K is either R or C. Furthermore let [ · , · ] be a Liebracket, that is, a mapping

[ · , · ] : L×L → L

such that for all A,B ,C ∈ L and a,b ∈K

1. [a A+bB ,C ] = a[A,C ]+b[B ,C ],

2. [A,B ] =−[B , A],

3. [A, [B ,C ]]+ [B , [C , A]]+ [C , [A,B ]] = 0.

THEN(L, [ · , · ]

)is a Lie algebra.

LET G be a n-dimensional C∞ manifold such that

• G is a group,

• the mapping ( · , · ) : G ×G →G , (g ,h) 7→ g h−1 is a C∞ mapping

• there is a countable covering of G consisting of open subsets of G .

THEN G is a Lie group.

A.1.2 Spherical coordinates

Let x = (x1, x2, x3) = (r cos(φ)cos(θ),r sin(φ)cos(θ),r sin(θ)) ∈R3. Then x = (r,θ,φ) ∈R3 de-note the spherical coordinates, where φ ∈ [0,2π], θ ∈ [0,π] and r ∈R+.

165

166 APPENDIX A. THE APPENDIX

A.1.3 Spaces of Lebesque measurable funtions

Definition A.1.1Two functions f , g :Rn →C are said to be equivalent almost everywhere, if they are equalexcept for a set of Lebesgue measure zero. (See [41] for the definition of Lebesgue measure.) Let1 ≤ p <∞. The spaces of Lebesgue measurable functions are

Lp (Rn) =

f :Rn →C : f is Lebesgue measurable and(∫Rn

| f (x)|p d x)1/p <∞

.

Here f is considered not as a single function but rather as the equivalence class of functions

which coincide almost everywhere. In this context ‖ f ‖Lp =(∫Rn | f (x)|p d x

)1/pis a norm for

Lp (Rn). L2(Rn) is a Hilbert space with scalar product ⟨ f , g ⟩ := ∫Rn f (x)g (x)d x.

A.1.4 Taylor’s theorem

Definition A.1.2 (Taylor polynomial)LetΩ⊆Rn , a ∈Ω and f ∈C k (Ω). The k-th Taylor polynomial is defined as

Tk f (x; a) :=k∑

l=0

1

l !

∑α∈Nn

0|α|=l

∂α f (a)(x −a)α.

Theorem A.1.3 (Taylors theorem)LetΩ⊆Rn , a ∈Ω and f ∈C k (Ω).

Then

f (x) = Tk f (x; a)+o(|x −a|k ),

as x → a.

That is,

limx→a

f (x)−Tp f (x; a)

|x −a|k = 0.

A.1.5 The formula of Faá di Bruno

Lemma A.1.4 (The formula of Faá di Bruno)Let g ,h ∈C m(R). Then

d m

d t m g (h(t )) = ∑b1,...,bm∈Tm

m!∏mk=1 bk !

g (∑m

k=1 bk )(h(t )) m∏

k=1

(h(k)(t )

k !

)bk

,

where Tm ⊂Nm0 is the set of sets nonnegative integers such that m =∑m

k=1 kbk .

A.1. CONVENTIONS AND DEFINITIONS 167

(For example

T1 = 1,

T2 = 1,0, 0,1,

since 2 = 2 ·1+0 ·2 = 0 ·1+1 ·2,

T3 = 3,0,0, 1,1,0, 0,0,1,

since 3 = 3 ·1 = 1 ·1+1 ·2 = 1 ·3

T4 = 4,0,0,0, 2,1,0,0, 1,0,1,0, 0,0,0,1,

since 4 = 4 ·1 = 2 ·1+1 ·2 = 1 ·1+0 ·2+1 ·3 = 1 ·4.)

Proof. For a proof see [28].

A.1.6 Multiresolution Analysis (MRA)

Wavelet systems which constitute an orthonormal basis of L2(Rn) are usually constructedstarting from a multiresolution analysis.

Definition A.1.5 (Multiresolution analysis (MRA))A multiresolution analysis (MRA) is a sequence

V j

j∈Z of closed subspaces of L2(Rn) such

that

1. V j ⊂V j+1, ∀ j ∈Z;

2. span(⋃

j∈ZV j)= L2(Rn);

3.⋂

j∈ZV j = 0;

4. f ∈V j ⇔ D− j2 f ∈V0;

5. f ∈V0 ⇔ Tm f ∈V0, ∀m ∈Zn ;

6. there exists φ ∈ V0 such thatTmφ

m∈Zn is an orthonormal basis in V0. φ is called a

scaling function.

Let W0 ⊂ V1 be determined by V1 = V0 ⊕W0. A mother wavelet ψ is then a function such thatTmψ

m∈Zn is an orthonormal basis of W0.


A.2 Distributions

A.2.1 Test functions

Definition A.2.1 (Test functions)

(S ) We define the Schwartz class of rapidly decreasing functions S (Rn) as the class of func-tions f ∈C∞(Rn), n ∈N, such that for each j ,k ∈N0 there exists a constant C j ,k ∈R+ sothat

ρ j ,k ( f ) = supx∈Rn , |α|≤ j

(1+|x|)k |∂α f (x)| =C j ,k <∞,

where α = (α1, . . . ,αn) ∈Nn0 is a multi-index. As usual we denote the partial derivative

with respect to α by

∂α f (x) = ∂|α|

∂α1 x1 · · ·∂αn xnf (x),

where |α| =∑nl=1αl .

(E ) E (Rn) :=C∞(Rn) where the topology is given by the set of seminorms

‖φ‖N := sup|x|≤N ,|α|≤N

|∂αφ(x)|.

(D) The space of test functions D(Rn) is given as the set of functions in C∞(Rn) that are ofcompact support. For a definition of a topology on D(Rn) see for example [40] Chapter6.2. A sequence φk k∈N ⊂ D(Rn) converges to φ ∈ D(Rn) iff there exists a compact setK ⊂Rn such that supp(φk ) ∈ K , ∀k ∈N and limk→∞ ‖φk −φ‖n = 0, ∀n ∈N.

Theorem A.2.2S (Rn) is a Fréchet space when endowed with the set of seminorms ρα,k .

S (Rn) is a dense subspace of Lp , 1 ≤ p <∞. D(Rn) is a dense subspace of S (Rn) which againis a dense subspace of E (Rn).

Proof. The proof of the fact that S (Rn) is a Fréchet space can be found in [40] Theorem 7.4.

The space of test functions D(Rn), is dense in S (Rn). (See [40] 7.10) But D(Rn) is dense inLp (Rn), 1 ≤ p <∞. (See [56] Lemma V.1.9.)

A.2.2 Distributions

Definition A.2.3 (Distributions)

(S ′(Rn)) Elements of the topological dual space S ′(Rn) of S (Rn) are called tempered distribu-tions.A function f : S (Rn) → K is an element of S ′(Rn) iff f is linear and continuous. Alinear function f : S (Rn) →K is continuous iff

∃C > 0, j ,k ∈N0 : |g (σ)| ≤C sup|α|≤k,x∈Rn

(1+|x|) j |Dασ(x)|, ∀σ ∈S (Rn).

A.2. DISTRIBUTIONS 169

(E ′(Rn)) Elements of the dual space E ′(Rn) of E (Rn) are called distributions of compact support.

A function f : E (Rn) → K is element of E ′(Rn) iff f is linear and continuous.A linear function f : E (Rn) →K is continuous iff

∃N ∈N0,C > 0 : | f (η)| ≤C sup|x|≤N ,|α|≤N

|∂αη(x)|, ∀η ∈ E (Rn).

(D′(Rn)) A function f : D(Rn) → K is element of D′(Rn) iff f is linear and continuous.A linear function f : D(Rn) →K is continuous iff for every compact set K ⊂Rn

∃N ∈N0,C > 0 : | f (η)| ≤C supx∈K ,|α|≤N

|∂αη(x)|, ∀η ∈D(Rn) : supp(η) ⊂ K , (A.1)

or equivalently iff for every convergent sequence φk ∈N ⊂ D(Rn) the sequence f (φk )k∈N ⊂K converges.

The order N of the distribution f is the smallest number N ∈N that will satisfy (A.1) forall compact sets K ⊆Rn . If no such number exists, f is said to have infinite order.

Definition A.2.4 (Support of a distribution and convolution with test functions)The support of f is defined as

supp( f ) :=⋂K = K ⊂Rn : supp(φ) ⊆ K c ⇒⟨ f ,φ⟩ = 0.

The convolution of a distribution f with a corresponding test function φ is defined as

f ∗φ(y) := f (Ty φ),

where φ(x) =φ(−x), ∀x ∈ (Rn).

Theorem A.2.5 (Convolution with test functions )

(i) Let f ∈D′(Rn), φ ∈D(Rn), then f ∗φ ∈ E (Rn).

(ii) Let f ∈ E ′(Rn), φ ∈D(Rn), then f ∗φ ∈D(Rn).

(iii) Let f ∈S ′(Rn), φ ∈S (Rn), then f ∗φ ∈ E (Rn), and

∂α( f ∗φ) = ∂α f ∗φ= f ∗∂αφ.

Moreover f ∗φ ∈S ′(Rn).

(iv) Let f , g ∈S ′(Rn) : f |D(Rn ) = g |D(Rn ) ⇒ f = g

Proof. (i) For a proof see [40] Theorem 6.30 (b).

(ii) For a proof see [40] Theorem 6.35 (c).

(iii) For a proof see [40] Theorem 7.19 (a), (b).

(iv) Let φ ∈S (Rn) ⇒∃φl l∈N ⊂D(Rn) :φl →S (Rn )

φ. Then

f (φ) = f ( liml→∞

φl ) = liml→∞

f (φl ) = liml→∞

g (φl ) = g (φ).


A.2.3 The direct product of distributions

Definition A.2.6 (The direct product of distributions)Let n,m ∈N, f ∈ V ′(Rn), g ∈ V ′(Rm), where V =D or V =S . The direct product of f and g isdefined as

( f × g )(φ) := f(g(φ( · f , · g )

)), ∀φ ∈V (Rn ×Rm).

Theorem A.2.7 (The direct product)Let l ,n,m ∈N, f ∈V ′(Rn), g ∈V ′(Rm),h ∈V (Rl ).

(i) The direct product given in Definition A.2.6 is well defined.

(ii) The direct product is commutative, i.e., f × g = g × f .

(iii) The direct product is associative, that is,

( f × g )×h = f × (g ×h).

Proof. See [58] Theorem 5.2-1, 5.2-2, 5.3-2 and 5.3-3.

A.2.4 Convolution of distributions

Definition A.2.8 (Convolution of distributions)Let f , g ∈V ′(Rn),φ ∈V , where V =D or V =S . The convolution of f and g is defined as

f ∗ g (φ) := f × g(φ( · f + · g )

)= f(g(φ( · f + · g )

)).

Theorem A.2.9 (Convolution of distributions)Let f , g ∈ D′(Rn) The convolution defined in Definition 1.3.8 is well defined if at least one ofthe distributions has compact support.

Furthermore, if f has compact support and ∃g ∈S ′(Rn) : g |D(Rn ) = g then

∃!T ∈S ′(Rn) : T |D(Rn ) = f ∗ g . (A.2)

Proof. We will first proof the result for f , g ∈ D′(Rn). Then we will show that |⟨ f ∗ g ,φ⟩| isbounded ∀φ ∈ D(Rn) with respect to the set of seminorms on S . From this it easily followsthat there is an continuous extension of f ∗ g on S .

Let f , g ∈ D′(Rn) such that supp f is compact and let h ∈ D′(Rn) such that there existsh ∈S ′(Rn) : h|D′(Rn )(Rn ) = h. Furthermore let φ ∈D(Rn), and set φ(x) :=φ(−x).

g ∗ f (φ) = g(

f(φ( · f + · g )

))= g((

f(φ( · f − · g )

))∨)(A.3)

= g ( f ∗ φ)∨.

By Theorem A.2.5(ii), ψ := f ∗φ( · ) = f (T · φ) ∈ D(Rn). Whence g ( f ∗ φ) is well defined, im-plying that g ∗ f is well defined.

A similar calculation shows that f ∗ g (φ) = f (g ∗ φ)∨. Furthermore, by Theorem A.2.5(i),g ∗φ ∈ E (Rn), and hence f ∗ g is well defined.


Moreover, f ∗ g = g ∗ f since the direct product is commutative (Theorem A.2.7(ii)).

Let us consider h ∈S ′(Rn). It is true that f ∗h(φ) = h(ψ) = h(ψ) = ˇh(ψ) and hence by Defini-tion A.2.3 we get

∃a > 0, j ,k ∈N0 : | ˇh(ψ)| ≤ a sup|α|≤k,x∈Rn

(1+|x|) j |Dαψ(x)|. (A.4)

Moreover, g ∈D′(Rn) and hence for every compact set supp(φ) ⊆ K ⊂ (Rn)

∃N ∈N0,b > 0 : |g (φ)| ≤ b supx∈K ,|α|≤N

|∂αφ(x)|. (A.5)

Now, by Theorem A.2.5(iii), ∂βψ= f ∗∂βφ= f (T · ∂βφ).

Since f has compact support, ∃ f ∈ E ′(Rn) : f |D(Rn ) = f . That is, f is linear and continuous onE (Rn) and thus by Definition A.2.3,

∃N ∈N0,C > 0 : | f (η)| ≤C sup|x|≤N ,|α|≤N

|∂αη(x)|∀η ∈ E (Rn).

Hence|∂βψ(x)| = |∂β f (Tx φ)| ≤C sup

|y |≤N ,|α|≤N|∂α+βφ(y −x)| (A.6)

Combined with (A.5) we obtain that for all compact sets supp(φ) ⊆ K ⊂ Rn and for all φ ∈D(Rn)

∃bK ,φ,Cφ > 0,kK ,φ, Nφ ∈N0 : | f ∗ g (φ)| = g (ψ)| ≤ b sup|β|≤k,x∈K

C sup|y |≤N ,|α|≤N

(∂α+βφ(y −x)).

Let M ⊂Rn be compact. Since BN (0) is compact for any N ∈N0 we know that there exists acompact subset K ⊂Rn and an N ∈N0 such that u = y − x ∈ M , ∀|y | ≤ N , x ∈ K . Thus for anycompact subset M ⊂Rn and for any φ ∈D(Rn)

∃c > 0, m ∈N0 : | f ∗ g (φ)| ≤ c supu∈M ,‖α|≤m

|∂αφ(u)|.

Hence we proved that f ∗ g ∈D′(Rn).

Combining (A.4) and (A.6), ∃a,b > 0, j ,k, N ∈N0 such that

| f ∗h(φ)| = |h(ψ)| ≤ a supx∈Rn ,|β|≤k

(1+|x|) j (C sup|y |≤N ,|α|≤N

|∂α+βφ(y −x)|)

Let u = y −x. Then |x| = |y −u| ≤ N +|u|. Hence for all φ ∈D(Rn)

∃C > 0, l , j ∈N0 : | f ∗h(φ)| ≤C (1+|u|) j supu∈(Rn ),|α|≤l

|∂αφ(u)|. (A.7)

Now we can show that ∃T ∈ S ′(Rn) : T |D′(Rn ) = f ∗ h. Let σ ∈ S (Rn). Then there existsφl l∈N ⊂ D′(Rn) such that φl →

Sσ. Since φl is a Cauchy sequence, by (A.7) f ∗h(φl ) is

a Cauchy sequence inK, that is, ∃T (σ) := liml→∞ f ∗h(φl ). Obviously, T is a linear extensionof f ∗h on S (Rn). Furthermore, by (A.7), T ∈S ′(Rn).

It remains to show that T ∈S ′(Rn) is the unique extension of f ∗h. To show this, let us assumethat ∃U ∈S ′(Rn) : U |D(Rn ) = f ∗h. Let σ,φl as above. Then

U (σ) =U ( liml→∞

(φl ) = liml→∞

f ∗h(φl ) = T (σ).


A.2.5 Products and distributions

Theorem A.2.10 (Products and distributions)

(i) Let g ∈S ′(Rn) and let f ∈C∞(Rn) such that

∀α= (α1, . . . ,αn) ∈Nn0 ∃kα,Cα : |(∂α f )(x)| ≤Cα(1+|x|)kα , ∀x ∈Rn .

Then the product f g ∈S ′(Rn) is well defined by setting

f g (φ) = g ( f φ), ∀φ ∈S (Rn).

(ii) Let g ∈D′(Rn). Then the product given by f g (φ) := g ( f φ) is well defined.

Proof. ad (i) The product is well defined if f φ ∈ S , ∀φ ∈ S . But this is clearly the case sinceany derivative of f is dominated by a polynomial.

ad (ii) The product is well defined since ∀φ ∈D(Rn) ⇒ f φ ∈D(Rn).

A.2.6 Fourier transforms of distributions

Recall that in Definition 1.3.11 the Fourier transform of a function f ∈S (Rn) was defined by

F ( f )(ξ) = f (ξ) :=∫Rn

f (x)e−2πi x·ξ =∫Rn

f (x)e−2πi ⟨x,ξ⟩Rn .

The Fourier transform F : S ′(Rn) → S ′(Rn) of a tempered distribution f ∈ S ′(Rn) was de-fined by f (φ) = f (φ), ∀φ ∈S (Rn).

Theorem 1.3.12 stated that the Fourier transform is an isomorphism of S (Rn). The inverseFourier transform is given by

F−1( f )(x) :=∫Rn

f (ξ)e2πi x·ξ, ∀ f ∈S ′(Rn), x ∈Rn .

The Fourier transform is a isomorphism of S ′(Rn). Its inverse on S ′(Rn) is given by

F−1( f )(φ) = f (F−1φ), ∀ f ∈S ′(Rn), φ ∈S (Rn).

It follows that

f (φ) = (F−1F f )(φ) =F ( f )(F−1φ) = f (φ∨), ∀ f ∈S ′(Rn), φ ∈S (Rn).


A.2.7 A Paley-Wiener theorem

Theorem A.2.11 (Paley-Wiener)

(i) Let u ∈D′(Rn) such that supp(u) ⊆ Br (0), and the order of u is N . (The order of a distri-bution was defined in Definition A.2.3.) Furthermore, let

f (z) := u(e−2πz ), ∀t ∈Cn , (A.8)

Then f is entire. The restriction of f toRn is the Fourier transform of u and there existsa constant 0 < γ<∞ such that

| f (z)| ≤ γ(1+|z|)N erℑ(z), ∀z ∈Cn . (A.9)

(ii) Conversely, let f be an entire function inCn which satisfies (A.9) for some 0 < γ<∞.

Then there exists u ∈D′(Rn) with supp(u) ⊆ Br (0), and such that (A.8) holds.

Proof. The proof can be found in [40] Theorem 7.23.

Corollary A.2.12The Fourier transform maps E ′(Rn) to E (Rn). The Fourier transform of a distribution f ∈E ′(Rn) is given by

F ( f )(ξ) = f (e−2πi ⟨·,ξ⟩), ∀ξ ∈Rn

Corollary A.2.13 (Product of tempered distributions)Let f , g ∈ S ′(Rn) such that f is compactly supported in the Fourier domain. Then f g ∈S ′(Rn) is well defined.

Proof. Note that the statement is a simple corollary of Theorem A.2.10(i). That the necessaryconditions are fulfilled is stated in Theorem A.2.11.

A.2.8 The Fourier transform and convolution of distributions

Theorem A.2.14Let f , g ∈D′(Rn) be such that f has compact support and ∃g ∈S ′(Rn) such that g |D(Rn ) = g .

Then f ∗ g = F g , where F ∈ E (Rn) is given by F (x) = f (e−2πi ⟨·,x⟩) and f ∗ g ∈ S ′(Rn) is theunique extension of f ∗ g given in Theorem A.2.9.

Proof. Let φ ∈D(Rn) ⊂S (Rn). Then

F 2( f ∗ g )(φ) = f ∗ g (φ) = g ∗ f (φ)(A.3)= g ( f ∗φ)∨ = g ( f ∗φ)∨. (A.10)

We will now look at the special case, where [ f ](φ) = ∫Rn f (x)φ(x)d x is a regular distribution

and f ∈D(Rn).


It follows that

([ f ]∗φ)∨ = ([ f ](T · φ)

)∨ = [ f ](T− · φ) = ˇ[ f ](T · φ) = ˇ[ f ]∗ φ= [ f ]∗ φ= f ∗ φ

=F ( f φ) ∈S (Rn).

Hence, using Theorem A.2.10(ii)

F 2(â[ f ]∗ g )(φ) = (â[ f ]∗ g )∨(φ) = (â[ f ]∗ g )(φ)(A.3)= g

(([ f ]∗φ)∨

)= g ( f φ) = g ( f φ) = f g (φ)

=F ( f g )(φ), ∀φ ∈D(Rn).

By Theorem A.2.5(iv)∀φ ∈S (Rn) : F 2(â[ f ]∗ g )(φ) =F ( f g )(φ). (A.11)

Now let f ∈D′(Rn) have compact support and let φ ∈D(Rn). If we define

[φ] ∈D′(Rn), [φ](η) :=∫Rn

φ(x)η(x)d x, ∀η ∈D(Rn),

then by Theorem A.2.7(iii)

[φ]∗ ( f ∗ g ) = ([φ]∗ f )∗ g ∈D′(Rn).

By Theorem A.2.9 ∃f ∗ g ∈S ′(Rn) whence ∃ ã[φ]∗ ( f ∗ g ) ∈S ′(Rn). Furthermore, since [φ] andf have compact support, so does f ∗ [φ]. Hence, by Theorem A.2.9, there exists an extensionã([φ]∗ f )∗ g ∈S ′(Rn).

Now Theorem A.2.5(iv) implies

[φ]∗ ( f ∗ g ) = ([φ]∗ f )∗ g ∈S ′(Rn). (A.12)

We first consider the left side of the equation (A.12):

(A.11) shows that

F ( ã[φ]∗ ( f ∗ g )) = φf ∗ g (A.13)

Now we consider the right hand side of the equation (A.12):

We can apply (A.11) to get

F ( ã([φ]∗ f )∗ g ) = àf ∗ [φ] g , . (A.14)

which, when simplified is

f ∗ [φ](ψ) = f(∫

φ(y)ψ( · + y)d y) u= · +y= f

(∫T · φ(u)ψ(u)du

)= f(∫

Tuφ( · )ψ(u)du)

= [ f ∗φ](ψ), ∀ψ ∈D(Rn).

Let τ ∈ D(Rn) : τ(x) := 1, ∀x ∈ supp( f ). Then by Theorem A.2.5(ii) f ∗φ ∈ D(Rn) and hence[ f ∗φ] ∈ E ′(Rn). It follows from Corollary A.2.12, that F ( f ∗φ) ∈ E (Rn). Whence

f ∗φ(y) = f ∗φ(e−2πi ⟨ · ,y⟩) =∫

e−2πi ⟨x,y⟩ f (Tx φ)d x = f(∫

e−2π⟨x,y⟩Tx φd x)

u=x− ·= f(τ

∫e−2π⟨( · +u),y⟩φ(u)du

)= f (τ

∫e−2πi ⟨x,y⟩Tx φd x)

= f (τe−2πi ⟨ · ,y⟩∫Rn

e−2πi ⟨u,y⟩φ(u)du) = φ(y) f (e−2πi ⟨y, · ⟩)

= φF (y). (A.15)


Combining (A.14), (A.15) and (A.13) we get

∀φ ∈D(Rn) : φ(f ∗ g ) = φF g ∈S ′(Rn). (A.16)

From (A.16) we now deduce our statement. Let us assume that

∃ψ ∈D(Rn) : F (f ∗ g )(ψ) 6= F g (ψ).

Then∃a > 0 : supp(ψ) ⊆Q := x ∈Rn : |x j | ≤ a, ∀1 ≤ j ≤ n.

Let Q∗ := x ∈Rn : |x j | ≤ b := π2a and let Q∗c

ε := x ∈Rn : Øv ∈ Bε(0), x + v ∈Q∗ for some ε> 0.Furthermore, let

φ ∈D(Rn) :φ(x) =

1, ∀x ∈Q∗

0, ∀x ∈Q∗cε .

Then φ(x) = ∫Q∗ e−2πi x yφ(y)d y +∫

Q∗ε\Q∗ e−2πi x yφ(y)d y =: I1 + I2.

I1 =n∏

l=1

∫ b

−be−2πi xl t d t =

n∏l=1

2sin(xl b)

xl

Hence, using Taylor series

|I1| ≥ (2b)nn∏

l=1(1− |xl b|2

3!) ≥ (2b)n(1− π2

223!)n > (2b)n2−n .

Now we can choose ε> 0 : |I2| < (2b)n2−n−1 to conclude

φ(x) 6= 0, ∀x ∈Q.

It follows that ∃Ψ ∈D(Rn) :ψ= φΨ and hence

F ( f ∗ g )(ψ) =F ( f ∗ g )(φΨ) = (φF ( f ∗ g )

)(Ψ)

(A.16)= (φF g )(Ψ) = (F g )(ψ).

which contraicts our assumption. Hence F (f ∗ g )(ψ) = F g (ψ) holds ∀ψ ∈ D(Rn) and henceby Theorem A.2.5(iv) ∀ψ ∈S (Rn).

List of Figures

1.1 Interrelation of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Example of steerable filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Example of an analytical signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Phase of the function f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Steerability of the Fourier multiplier of the Riesz transform . . . . . . . . . . . . . 26

3.1 Phase and phase direction of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Phase and phase direction of the Riesz transform . . . . . . . . . . . . . . . . . . . 69

5.1 Commutative diagram for the Riesz and wavelet transform . . . . . . . . . . . . . 79

5.2 The functions p0, . . . , p5, f6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 The function h corresponding to p0, . . . , p5, f6 . . . . . . . . . . . . . . . . . . . . . 97

5.4 Wavelet for L2(R2) in the case m = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5 Example of a monogenic wavelet decomposition . . . . . . . . . . . . . . . . . . . 103

5.6 Fourier filters of the monogenic wavelet . . . . . . . . . . . . . . . . . . . . . . . . 104

5.7 Filter bank for decomposition and reconstruction of a signal by the monogenicwavelet frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.8 Equalization of brightness of an artificially corrupted image . . . . . . . . . . . . . 107

5.9 The optical illusion of injecting gray disks is reproduced by Equalization of Bright-ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.10 Adelsons checkerboard illusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.11 Example of a monogenic wavelet decomposition . . . . . . . . . . . . . . . . . . . 109

7.1 The Gibbs phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.1 The basis elements of 2−D spherical harmonics . . . . . . . . . . . . . . . . . . . 157



177

178 LIST OF FIGURES

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Index

(·, ·), 44, 45, 50Bε(a) := x ∈Rn : À3AF x −aÀ3AF ≤ ε,Cν

k Gegenbauer polynomials, 139C f Cauchy integral, 73D Dirac operator, 58Dd isotropic dilation, 11H p (Rn+1+ ) Hardy space, 70Hk (Rn), 140Hn Weyl-Heisenberg group, 112L(·, ·), 52Lp (Rn), 166Mb modulation, 11Pk Legendre polynomials, 139P m

k associated Legendre polynomials, 159Px0 Poisson kernel, 23R Riesz transform, 20, 66Rα partial Riesz transform, 20, 33Rd Riesz transform in direction d , 27Sn−1 := x ∈Rn : À3AF xÀ3AF = 1,Sα, 54T synthesis operator, 80T H p

± (Rn), 74, 79, 88T ∗

adjoint of the operator T , 53analysis operator, 80

T ′ adjoint of the operator T , 52Tk Tchebichef polynomials, 139Tk f (x; a) Taylor polynomial, 166Tt translation, 11V0, 99W ∗ dual Clifford module, 47W k,p (Rn) Sobolev space, 11, 91, 114, 125W s (Rn) Bessel potential space, 11Wψ wavelet transform, 13, 87Dρ Wigner D-matrix, 143Ker(S) kernel of the operator S, 53Φ(Rn) Lizorkin space, 28Φ′(Rn) tempered distributions modulo poly-

nomials, 28

Ψ(Rn) , 28Ψa analytical wavelet, 79Ran(S) range of the operator S, 53ℜ(·) real part, 47À3AF ·À3AF modulus, 44⟨·⟩α, 42, 47⟨⟨·, ·⟩⟩, 45, 50~· vector part, 43Rp singular integral operator with kernel P ,

142E0(Rn), 28Ld , 34Wa,b,t , 112Hk (Rn) solid spherical harmonics, 138Pk homogenous polynomials of degree k, 138P(·) power set, 42· conjugation, 45, 50∂ Cauchy-Riemann operator, 58∂α, 8φ phase, 62, 69φ scaling function, 99ψm monogenic wavelet, 88exp exponential function , 62ρ rotation as operator , 11sgn(αβ), 42SO(n) := A ∈Rn,n : det(A) = 1,4 Laplace operator, 11, 138∧ Fourier transform, 9, 58a amplitude, 16, 62, 69d phase direction, 62, 69dWa,b,t , 112dn,k dimension of Hk (Rn), 139e0, 42eα, 42eα, 42fm monogenic signal, 68fa analytical signal, 16hn Heisenberg Algebra, 112u f Poisson transform, 23

183

184 INDEX

·† pseudo inverse, 79, 86E (Rn) , 168E ′(Rn) distributions of compact support, 169F Fourier transform, 9, 58H Hilbert transform, 15Hk (Rn) surface spherical harmonics, 138S (Rn) Schwartz space, 8, 168S ′(Rn) tempered distributions, 8, 169Cn complex Clifford algebra, 45D(Rn) test functions, 168M monogenic signal, 68Dd rotated dilation, 11On , 421, . . . ,n := α ∈N : 1 ≤α≤ n,D′(Rn) distributions, 169Rn Clifford algebra, 42lim-n.t. non-tangential limit, 73

a.e., see almost everywherealmost everywhere, 166amplitude a , 16, 62, 69analysis operator T ∗, 80analytical

signal fa , 16wavelet, 79

Bedrosian identity, 36Bessel

bound, 81sequence, 81

Bessel potential space W s (Rn), 11

Cauchy integral, 73Cauchy Riemann equations, 59, 71

generalized, 24Cauchy-Riemann operator ∂, 58Clifford

algebracomplexCn , 45

algebraRn , 42frame, 83functional, 47Hilbert module, 50module, 46

diffeomorphism, 92dilation

isotropic, 11rotated, 11

Dirac operator D , 58

distributionsD′(Rn), 169convolution

with a test function, 169convolution of , 9, 170direct product of , 8, 170of compact support

E ′(Rn), 169Fourier transform, 173

product of, 9, 172support, 169tempered

S ′(Rn), 8, 169Fourier transform F , 172modulo polynomialsΦ′(Rn), 28product, 173

division algebra, 43

Euler angles, 159Exponential function exp, 62extension principle, 98

formula of Faá di Bruno, 167Fourier transform F , 9, 58

and convolution of distributions, 173of distributions of compact support, 173of tempered distributions, 172

frame, 13, 92bounds, 13, 83, 92Clifford, 83decomposition, 13, 83, 86dual, 13, 83

canonical, 84, 86inequality, 13operator, 83tight, 13, 83, 95wavelet, see wavelet

Fueter variable, 61fundamental function of multiresolution, 99

Hahn-Banach theorem, 49Hardy space, 70Heisenberg algebra, 112Hilbert transform H , 15homogenous, 24, 138

infinitesimal operator, 112instantaneous frequency, 18, 69

κ-function, 99

INDEX 185

Laplace operator 4, 11, 138Lizorin spaceΦ(Rn), 28

monogenicsignal, 68wavelet ψm , 88function, 59

MRA, see multiresolution analysismultiresolution analysis, 167

Neumann series, 85non-tangential limit, 73

operatorleft-linear, 47, 53adjoint, 55

in a Clifford Banach module T ′, 52in a Clifford Hilbert module T ∗, 53

self-adjoint , 55unitary, 55

Paley-Wiener theorem, 173paravector, 43paravector group, 43partition of unity

regular, 93Riesz, 93, 97

phaseφ, 69φ of the analytical signal, 16angle φ, 62direction d , 62, 69

Plancherel inequality, 58Plemelj projection P±, 74Poisson

kernel Px0 , 23transform u f , 23

Poisson summation formula, 10polar coordinates, 63polynomials

Gegenbauer Cνk , 139

homogenous Pk , 138Legendre

associated P mk (x), 159

Legendre Pk , 139Taylor Tk f (x; a), 166Tchebichef Tk , 139ultraspherical Cν

k , 139power set P(·), 42pseudo inverse ·†, 79, 86

refinable function, see scaling functionrefinement mask H0, 99representation

equivalent, 143of a group, 14of an algebra, 14of the rotation group, 143–144reducible, 14unitary, 14

Rieszpartition of unity, see partition of unityrepresentation theorem, 52transform

higher order, 161transform , 20, 66

higher, 143in direction d : Rd , 27on the Lizorkin space, 30order of higher, 143of distributions, 33of tempered distributions, 33of tempered distributions modulo poly-

nomials, 32partial, 20

scaling function, 99Schrödinger representation, 112Schwartz space S (Rn), 8, 168Sobolev space

W 1,2, 114, 125W k,2, 91W k,p (Rn), 11

spherical coordinates, 63, 165spherical harmonics

realR2, 156R3, 159

solid Hk (Rn), 138surface Hk (Rn), 138

steerable, 27filter, 7, 162pyramid, 162

synthesis operator T , 80

Taylorpolynomial Tk f (x; a), 166theorem, 166

test functions D(Rn), 168

uncertainty relation

186 INDEX

for Riesz transforms, 125for self adjoint operators, 113Heisenberg, 113, 114, 123, 125

waveletanalytical, 79frame, 13, 79, 88

radial, 95tight, 94, 95

hypercomplex, 87mask, 99monogenic

frame, 88monogenic ψm , 88mother wavelet, 13, 87set, 99system, 13, 87transform Wψ, 13, 87

Weyl-Heisenberg group, 112Wigner D-matrix, 143, 156

zero divisor, 43

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